I've seen this issue before; my computer will appear to be running smoothly and then randomly shut off. It seems to last an average of about ten minutes. The first thought was that my power supply was dying, so I got another one and the same problem occurred. I've also tried switching out a different, lower power video card.
The power supply fan, CPU fan, and case fan all seem to be running normally. I've managed to run SpeedFan before my PC shuts off and it appears that Core0, 1, 2, 3 are running very hot.
What are the chances that it's my motherboard and/or CPU?
Any advice on the situation would be greatly appreciated.