Aha! I took mine apart and I see now. Yep, it's got some play if I move it a bit, so I'm guessing it could be the torque of the fan causing the parts to rub. Looks like I'm off to buy some superglue.
(I know I've bought superglue in the past, repeatedly, but every time a project comes up that needs it, I can't find any. Argh!)