Of course while I'm at the Apple Store (it's an hour drive to get there) the computer works perfectly. The "Genius Desk" dude ran some software diagnostics and everything checked out fine. He went through a procedure to reset the power module (or something like that), which entailed removing the battery and holding down the start button. He shut down the machine and tried starting it again probably ten or twelve times. It worked perfectly every time.
Naturally, I get home, go to fire it up... and nothing. Black screen again. It took me five tries to get it to start.