The trust is wrestling with an expensive problem, which it never intended to undertake: managing one of the largest museums in the country, and in ways that would cause any conventional museum heart failure.
If Stourhead were a purpose-built museum, you'd never be allowed to walk straight from the great outdoors into some of the most important galleries, full of treasures.
Instead, since the modern visitor approaches as if they were a guest of the vastly wealthy banking family, which flattened a Tudor mansion to build their severe Palladian palace, the tall front door swings open into the magnificent marble-floored hall admitting a gush of fresh air - though the dust from the famous gardens is almost removed by walking across three hideous mats.
The visitors are hit straight away by a tidal wave of ancestors, and must fight to keep afloat with remarkably little help. They might just know that the house is an exercise in early 18th-century good taste. They might just have heard of the Hoare family which built (and still partly occupies) Stourhead, but essentially they're on their own.
They might guess that the chap in hunting pinks, hanging over a gruesome fox mask under a glass dome, was given to a bit of recreational slaughter, but who are all these other stern men and spectacularly bejewelled women lining the walls of the great entrance hall?
There are no labels, no text panels. There is a picture list on the console table in the far corner, chatty volunteer room stewards, and a guide to the house on sale from the polished hall table. But if 72,000 visitors a year stopped in their tracks to read it immediately, nobody would get any further.
The next room is stuffed with Chippendale, one of the National Trust's most important furniture collections, made for the house, but oddly positioned because of the need to clear one side of the room as a visitor walkway: the tea drinkers would be jammed up against the fireside chair occupants, and in great danger of having their cups sent flying by anyone at the music stand.
The room did not exist in the time of Henry Hoare or Thomas Chippendale, but was added after a fire: again the visitor gets no clues except an Edwardian photograph showing the room even more stuffed, but with completely different furniture.
The inevitable silk rope keeps visitors off an Oriental carpet, which is beautiful but so worn it's hard to believe the Hoares, who for generations liked everything to be immaculate, would have given it house room.
Grasby, responsible for some spectacular National Trust collections in the south-west, sighs deeply and invites me to see the room as work in progress. Like many of the trust's curators, he spends his working days trying to balance the juggling act of being a keeper 'for everyone forever' of historic properties, which also hold a vast national museum.
The National Trust never intended to run a museum. It was set up in 1895 by three wealthy philanthropists concerned, that development and industrialisation were about to choke the green heart out of the English countryside.
The historic properties are still dwarfed by the land holdings, over 600,000 acres and over 700 miles of coastline. However, as the properties came in - this month sees the centenary of the acquisition of Barrington Court in Somerset, the trust's first major country house - so did the collections.
Last year the trust became the largest accredited museum authority, with 149 properties registered as museums, almost one in ten of all the registered museums in the country. From the folly cottage A La Ronde in Devon to Washington Old Hall in Tyne and Wear, if the entire collection were brought together it would burst the seams of the British Museum.
There are well over a million objects, from Georgian mantraps to the Oscar awarded to George Bernard Shaw for Pygmalion, from 1,000 Beatrix Potter drawings and watercolours to the spectacular 16th-century Guido Reni ceiling at Kingston Lacy, transplanted in the 19th century from a Renaissance palace in Bologna to a country house in Dorset.
Many of the collections vividly reflect the frequently eccentric personalities of their former owners. Hundred Wheels, a large attic at Snowshill, is full of the prams, bicycles, dog carts and model farm wagons assembled by Charles Wade: when the house had to be emptied for rewiring, a small army of volunteers and 55,000 sheets of tissue paper were assembled to pack what was understood to be 20,000 objects - eventually it turned out there were half as many again.
The collection is constantly growing. The trust has launched appeals, and won lottery grants, to buy back original contents. Recent properties, already some of their most popular, including Paul McCartney's old home in Liverpool, and the Birmingham back-to-backs, had lost almost all contents: the trust needed to buy or persuade somebody to give everything from enamel teapots to valve radios and green lino.
Valuable collections have also been donated or bequeathed that don't relate to any property, and can sometimes be matched with properties that have lost theirs.
In some properties over half the collections are on extended loan from their original owners, with obvious implications when future generations feel the blood ties loosen and the tax man beckon. Other properties come with an embarrassment of riches: one of the most spectacular acquisitions of recent years, Tyntesfield near Bristol, came with 40,000 objects.
The collections include both treasures of international importance, such as the Chippendale, and what could look like the sweepings of a junk shop. At Mr Straw's House in Worksop, without every scrubbing brush and every pill bottle hoarded by the owners, the building would be a bog-standard small town redbrick of no architectural or historical interest.
Curators are currently working through the rooms of Scotney Castle in Kent, a property the trust has owned since 1970, but only gained access to last year when Betty Hussey, the formidable widow of their benefactor Christopher Hussey - a former editor of Country Life, and notable historian of country houses, including many now owned by the trust - died aged 99.
In just over a year it has managed to open part of the ground floor to visitors. Years of work lie ahead on the ground floor, and all the upstairs rooms, including Betty's bedroom, which she specifically wanted opened to visitors, with instructions to any conservator that it should be displayed in full daylight with the blinds up, so visitors might share her delight in the spectacular view.
Other odder conservation decisions will have to be taken soon. Betty loved cats, which had the run of the house. On a side table in the library lies her copy of the May 2006 issue of Your Cat, which arrived just at the time of her death, and now touchingly waits for her to come in.
It is still in mint condition, but will clearly not remain that way: should they allow it to yellow and decay, buy a stack of back issues or continue the subscription so there is always an up-to-date copy waiting for her? The decision has not yet been made.
'In some ways it's wonderful to have a blank page here,' curator Neil Walters says. 'The answer has to be to find the best solution for each property rather than a one-size-fits-all approach. But I hope we can keep barriers to a minimum, and signs out of the picture completely.'
But because the collections exist to help tell the stories of the properties, which must make sense as historic interiors, the trust can't take the conventional museum approach either for display, interpretation, or conservation of extremely precious and acutely fragile objects.
In most rooms at most properties, text panels, wall labels, sealed climate controlled cases, lights dimmed to a glimmer, might make conservation and interpretation sense but would destroy the character of period interiors.
Sarah Staniforth, the head of historic properties, says that the National Trust's voluminous research proves that the winter closures, and exemplary housekeeping (their famous Manual of Housekeeping was published in 2005 in one massive tome) delivers collection care 'comparable or superior to' display in conventional museum cases.
Expertise is increasingly being shared with others such as English Heritage and Historic Royal Palaces, and through the hubs and work placements with independent and local museums.
The cost of caring for it all is terrifying: the estimated price tag of restoring the state bed of James II and its astounding hangings, which ended up at Knole in Kent, is £800,000.
The National Trust is only now getting to grips with this monster at a national level. At the end of this year a database of the entire collection, including the ice-skates that tumbled out of a cupboard at Scotney a few months ago, will start to go live for staff. The hope is within a few years to make it accessible to the public - although problems remain, chiefly over security, and copyright issues over the loan collections.
Interpretation is currently the subject of endless debate. 'This is one where we have to hold up our hands and say there is some way to go,' Staniforth says. The trust has made some cautious experiments with costumed interpreters, is being urged to adopt widespread use of audio guides, and has experimented at some properties with formal presentations by room stewards.
Conservation work is increasingly becoming part of the interpretation, carried out under the gaze of a generally fascinated public. Some properties are managing mini-museum displays within the properties but out of the main rooms: at Ham House in Surrey visitors learn of the mansion's torrid romantic and political past through a video and text panels in the former dairy.
James Grasby has moments when he wonders if his overriding obligation to preserve isn't actually fatal to visitors' understanding how the houses and the objects worked. He sometimes fears that the attempt literally to stop the rot, to freeze a house and its contents at a moment in time, creates a lie.
He would like visitors to see Colt Hoare's staggering Georgian library at Stourhead as the engine room driving the house and estate, as well as his own artistic, literary and archaeological interests.
If every visitor took a book from Hoare's shelves and sat on a Chippendale chair at his Chippendale desk, or opened the great leather bound volumes of his drawings, they might suddenly get it - but, with 72,000 pairs of hands a year, the destruction of the objects would be disastrously accelerated.
The effect could be achieved by putting the objects into store and displaying sacrificial replicas, but he feels that would destroy the unique and magic promise; the encounter with the real object. He wonders if part of the answer could be to persuade visitors to work harder before they arrive.
Could they be invited to download MP3 files to play on their way to the property? Could they watch in advance video clips of books and albums being opened, be sent book lists or audio files of readings from the works of literary residents? (He has had visitors turn up at TE Lawrence's Clouds Hill in leafy Dorset, expecting an encounter with the grim mining background of DH Lawrence.)
'What we have to do is find new ways to make a person's visit as rich as possible, a personal encounter with the spirit of the place, not just a quick wander round and then tea,' Grasby says. 'I do sometimes worry that we lose the whole flavour, in not being able to drink a deep enough draught of the past.'
Maev Kennedy is the heritage correspondent at the Guardian