Happens all the time. Modern paleontology/archaeology addresses human bones extremely thoroughly. In the past....well....not so much. If Bigfoot believers want to obtain the taxonomic and anatomical knowledge necessary to identify fossil mammals and sort through the collections in museums, I'm all for that. They won't find any Bigfoot remains (I seriously doubt that we'd have extracted ALL the bones), but they'll certainly make other significant discoveries. I don't believe we've classified much of the stuff from the Bone Wars; it's saurian, not mammalian, but it illustrates my point.
Still, your characterization isn't accurate. Even in the past human remains were fairly well studied. No one "just toss[es fossils] in drawers like they're no big deal". Museum curation is a process, and involves much more than simply tossing stuff in drawers. The most poorly-studied specimens will be put into boxes, with a tag listing a unique identifying number, collector, location, lithology, date of collection, and a tentative identification. This tentative identification can be "Brachiopod", but it's there--and with ape bones of any kind, it's going to be much more, given the high interest in human evolution. If the specimen doesn't have such a tag, it's basically worthless (fires can render whole collections completely worthless, even if they don't damage a single specimen).
If you want to see what's involved, I recommend poking around in this Zooniverse project. It shows what's required to be on labels for museum specimens (along with the annoying degree of variation therein).
