Are you talking about the Tascam plastic reel clampers like those on the TSR-8, 38, etc?
Yes, those wear out and will fail to hold the reel firmly enough. It's best to put a Tascam machine on its back or at least angled back in a rack stand. I'm not sure about compatibility with other brands.
Wobbling could be something else, but the reel clampers are often the culprit. A good way to tell is to try to close them all the way with no reel mounted. If there is a gap where the two sections should meet they should be replaced.
-Tim