Many thanks to both of you. I have not yet received a reply from Customer Support, but am hopeful they are aware of the problem. I am now usually able to duplicate the bug, but still cannot solve it. Essentially, the character who has a given set, or pieces of that set, is the only one who will see the item in collections. And if that character isn't currently wearing the set (post-bound,) it says it is not in collections. Sometimes, if I put all or some of the items on the character who had the set first, the pieces will reappear in collections only while I'm logged into that character. But that information will only sometimes, but not every time, carry over to collections on another character (when it does, the entire set must be equipped on the first toon, as in I can't take them back off whomever owns them without them going missing from collections,) and sometimes, even when I'm actually wearing the entire set, it still says I own 0/0, even on the character wearing them.
I'm very much hoping this is just a display bug, and that I have not lost anything, but I cannot currently claim any affected armor set from collections.