Computer turns on for a split second -- then nothing

Quick history....

Computer was acting strange. Video seemed very odd and flakey. I replaced the video card and it seemed to be working fine.

About a week later, now the computer is dead. If I press power, the fans spin for about .25 seconds and then stop. There are no beeps, no error codes, no nothing. The power button on the the motherboard and the CMOS are illuminated but those are the only signs of life.

This hasn't happened to me before. Where do I begin to troubleshoot? Motherboard? Power supply? Would a blown power supply spin the fans at all?

Thanks for any guidance you can provide.

Yes, a blown PSU could spin the fans for a moment.

Could be the mother board is touching the case and shorting out. You could try "breadboarding" it. Remove the motherboard, place it on cardboard, install psu and see if it will boot.
Touch a screwdrive on the power switch connector on the motherboard to start.

Or if you have a spare PSU around try that first.