Well, there wasn't anything obviously touching any fans, but when I powered back up, the main case fan wouldn't start, which is all kinds of bad. Also, the reboot indicated that one of the mirrored hard drives had failed, which is all kinds of worse.
A bit of fiddling with the case fan got it running again. The hard drive, sadly, is almost certainly toast.
So I've gone to NewEgg and ordered a replacement hard drive and a new case fan that should be here soon.
And I'm backing up changed files to the NAS. (That's something I should have done anyway, but it's now become urgent.)
Eventually, I will have replaced all of the parts in this box...