I had a similar but slightly different problem once. My computer kept switching off as though the power cable was being pulled out. Monitor, Printer, CPU. Every now and again it just went.

As it turned out what was happening was I was pushing the desk back onto the multi-socket adaptor which has a switch on it which cuts the power when flicked. I don't think that helps you though.

I dont know much about this stuff, but if your video card blew then I think your computer would use its very basic card and would still switch on in safety mode. Could it be the Power Supply? I had to get a new one when I got a new Video card because the power rating on my old one was too low.

http://en.wikipedia.org/wiki/Computer_power_supply

I think they are pretty cheap so it could be worth buying one and seeing if it works.