Yes, very possible. Versions of lost plays and other Hellenistic writings have been found inside the body cavities of mummies, used as packing material during the embalming process. The most famous caches of lost documents were both hidden by religious groups: the Dead Sea Scrolls, the earliest known copies of Hebrew scriptures, and several libraries of Gnostic texts preserving early versions of the Gospels have been discovered.

If such texts are discovered, they are most likely to be found in dessert regions like Egypt and Israel; the arid climate greatly assisted in preservation of the papyrus. The texts that are found this way are generally ones whose existence historians were already aware of, since only the most popular of texts would be copied in enough profusion to be at hand when an embalmer needed it, and didn't care enough about it not to use it for that purpose. As for materials hidden away in undisturbed caches, those have generally be religious in nature, secreted by cults or other minority religious movements.

Texts can be found anywhere, not necessarily in far flung or exotic locations like a secret Vatican library or a tomb.

For instance, many ancient texts were repurposed - they were scraped off and recopied as palimpsests or reused to bind other books that were thought to be more important. Sometimes, these texts are rediscovered when someone needs to work on the binding of an old book, or someone just notices some interesting scratching and moves to look at a text with technology that will reveal some earlier writing in a palimpsest.

The important thing to realize is that this writing will be very fragmentary - no one is digging up full books. That is extremely rare and unlikely, given the nature of ancient/medieval book production (i.e. even having a "full" book of the Bible was rare in the Middle Ages, when you'd think that kind of thing would be rather common).

Also, another reason why it is very likely that there are probably the words of some precious text hidden somewhere in plain sight is that there are really a lot of books from the Middle Ages (that's were almost all our ancient books come from; very very very little is extant from the ancient world). And while there is a huge haystack, there are very very few archivists looking for these things, and even fewer with the expertise and financial backing to initiate a project to discover something like this even if they thought it might be there (if they didn't have strong institutional backing, it might take years to get a research project like this off the ground).

So the long and short of it is that archival work is the place your likely to find new texts, if new texts are found they are likely to seem unimpressive to the general public, and there just aren't enough archivists and resources for them to find whatever shreds might be there.

Ok, that's the reality of the situation. If I were to dream, it would turn out that the Cottonian Library was in fact mostly saved and has been entombed beneath the Palace of Westminster for hundreds of years, only to be accidentally discovered during a routine bathroom renovation or something.

I can only address a sliver of your question, specifically the portion regarding the Vatican archives. The Vatican archives are much less of a mystery than you might think. They are fairly standardized in their operation, including the "Vatican Secret Archives." It's not that there are "secrets" in there, it's the definition of "secret" that means private. (Think "secretary" or "secret service") Basically, it's the pope's writings. There is a standard 75 year wait for the archives to be opened, and generally it goes from the beginning of a particular pontiff's papacy. Right now, we have access to the papers of the pontificates up to and including Pope Pius XI, who died in 1939. The release of the papers of Pope Pius XII, who took office in that same year, are slightly behind schedule. There would be no ancient texts in the documents from his pontificate, however.