Yes, it seems like a normal boot with the beep and all, there's just no display on the monitors. Turning off one of the monitors seems to have the same result. I should point out that this sometimes happens during warm reboots as well. It's kind of random. In fact, last night my main monitor had display during BIOS but the secondary monitor didn't until it suddenly showed display a few minutes after Windows was already booted.
The machine has an Antec NeoPower 480W PSU. I thought that would be powerful enough, but what you said about there not being enough power makes sense because I have a better chance of getting display if I quickly hit the power button off and on if the machine is already running.
Triniton: Yes, I have onboard video but it is disabled. My main card is PCI-e and "First Display Device" is configured that way.