Well, I lugged the laptop into BnA HQ today, thinking I'd be handing it over to coworker and computer guru Mika to salvage what he could from the hard drive before I shipped the whole thing off to Texas to be repaired... And then when he turned on the computer, it powered right up. Sluggishly, but it worked.
"Yeah," he said, "I sometimes have this effect on computers."
So I still have no idea what was wrong with it yesterday, but Mika scared it into submission today, and, while it was still working, he and I went to the office store, bought an external hard drive and did a full backup. Just in case. The machine is home now (with its new external hard drive happily camped out next to it), and seems to be fine. I'm not going to look a gift horse in the mouth, I'm just going to continue being really happy it's working and my data is now safe.
And again, let this be a lesson to you - back up your computers regularly. You don't want to give yourself an ulcer.