We've run diagnostics and tried resetting the PRam a few times, so odds are either Power Management's screwed up, or there's a backup battery out. It can't see its battery either, which is consistant with these potential reasons for its problems. Either way, I can't take it in for repairs until tomorrow - and of course, it's a month beyond the warranty period. Ah well. At least it seems otherwise in good shape.
In any event, this likely means I shall be slow to respond to mail and anything else online for at least the next few days. We'll see...