Well, I just learned something. My iPod stopped responding to the wheel or to clicking anything. So, I tried to reset it. This is Apple's fix-all for iPods. To reset it you click the hold switch on and then off, then you hold down the menu button and the center button until the Apple logo appears. This didn't help, nor did doing that while it was plugged in. It would reset, so it wasn't a hardware issue.
Eventually I, by chance, tried a reset when it was asleep. This fixed it right up and everything is now good.
EDIT -- Or not. I guess this was a fluke. Damn.