brett:
GeForce Graphiccards tend to need a great peak power to startup. So if you use a large CPU fan (something like a 80x80 fan) and you've got that attached to your mainboard the system may require more then 300Watts to boot successfully.
The real problem is, that all components need more power while powering up than during normal operations.
Otherwise, yes, a mainboard may die slowly due to heat or dust. A layer of dust can short a few circuits and cause the funniest symptoms in irregular patterns.
I had a server to fix, which lost the harddisks, cdroms and tape drives at random. The cause was a thick layer of dust on the mb which shortet a few circuits in the vicinity of the on-board SCSI controller.
I'd try the graphics card in a different system to check if it is not fried.
If it is still okay, try your system with a larger power supply and the geforce 4.
If that still doesn't work, change the motherboard.
HTH
Cheers,
Scotty