Major ick in the case, particularly around the fans. Cleared that out and am getting ready to see if it's made a difference. Fortunately, all four fans were running (one on power supply, one on motherboard, and two in the case). They are now, however, running more quietly, which can only be a good thing, right?

Although, if it were an issue with the video card overheating, wouldn't that happen after playing for a while or after the computer had been running for a while rather than happening after boot-up?

Oh Dear Lord, please don't make me have to replace the video card. The computer is my lifeline, and I've seen what good cards run.