Everything else being all right, it's entirely possible to bring the piano back up to pitch after it has sat untuned for a long time.
I've done it many times. As MMazzon said, stability will definitely be a problem at first -- the piano will go back out of tune a lot faster than normal. The main cause of this is that the wire forms bends around several bearing points on its path between the tuning pin at one end and the hitch pin at the other. When you drastically change the tension on that wire, those bends end up in a slightly different place. Over a period of time, the old bend will relax and a new one will form, causing the string to go out of tune. So in order to get the instrument stable again, you have to keep retuning it as soon as it goes out. A typical recommended schedule after an extreme pitch raise is something like retuning in one month, then two months, then three... It works. Will it ever be AS stable as if it had never been neglected in the first place? Maybe not, but it'll be pretty good. Certainly tuneable.
Keep in mind, the cost of all those tunings kinda adds up, and that's where perhaps the greatest truth of the statement about a piano becoming "untuneable" comes from. Usually the kind of owner that will let their piano go for a decade without tuning, usually isn't going to be willing to go to the trouble or expense of having it retuned again that often.
Now for the implicit question that I dodged earlier: will everything else be all right?
That's a lot harder to answer, and I would venture nobody really knows. There are too many factors and they're pretty hard to measure. Making large changes in tension on the strings certainly isn't great for a piano. It tends to do things like work the bridge pins loose, stress various glue joints, etc.. But to a degree, they're built strongly enough to handle some of this. Consider that it's common practice in rebuilding shops to completely remove all the strings, work on the belly, and then restring. Usually with new strings (might as well), but not always! It works fine, as long as everything was still solid to begin with. You're putting stress on the parts, but as long as they were healthy they can handle it.
Another thing that comes into play is rusty strings. If the strings are rusty and the piano has fallen way below pitch, it may not be possible to bring them back up to pitch without breaking them. Whereas if the piano had been kept at pitch, they might still survive a more minor adjustment.
With a typical modern piano (late 1800s on), there shouldn't be any issues with the frame changing shape. The plate and bracing is so massive and stiff that it just doesn't move much.
