Initially I thought it was a power supply problem, as the fans on the power supply wouldn't turn on. I assumed there was enough voltage to turn the light on, but not start the computer.
So I tried using a spare power supply. Still it didn't work. Uh oh, motherboard problem maybe....
Then, on a hunch, unplugged the machine, pulled the clock battery, hit the power button to discharge. Then plugged it in without the clock battery. The machine booted perfectly. I shut down, reinstalled the clock battery, and now the system is fully functional again.

