Thanks to EVERYONE for your helpful responses.

The shop it was at told me I needed a new fuel pump to the tune of $320, and then (seriously) tried to hold my car hostage. But I was armed with the glorious wisdom of brickboard.com and informed him I'd be taking it away now.

My own mechanic isn't open, but a mechanically-minded friend of a friend came over, and with his common-sense and me reading out snippets of these comments, he diagnosed the problem in 15 minutes:

When my car was serviced, one of the battery leads (the one with the little wire that goes to the alternator) hadn't been tightened properly. Going over the speed bump jarred it out of contact, the tow guy's clamping-on of the battery pack put it back in contact.

The friend of a friend tightened the bolts, started and shut off the engine a half-dozen times to be sure this wasn't blind luck, and I just drove around the block for ten minutes to be extra sure. Problem fixed. I even went over a few bumps just to check. ;)

THANK YOU, brickboard folks! Without the confidence I'm (very slowly) gaining from lurking in these forums, and your answers, I never would have dared contradict A Professional Mechanic, and I'd be 300+ bucks poorer.