First thing that you can check is to go into your BIOS when the computer starts and see what the voltage outputs of the power supply are. You'll find this on the H/W monitor tab. If the the voltages look too far off this is probably your problem.
Next thing that you can try is running a burn-in program on your system. You can use something like SiSoft Sandra http://www.sisoftware.net/ for this, you can download a free version. If your computer shuts down, it's probably the power supply.
Next thing you can try is a graghics benchmarking program such as 3DMark05 http://www.futuremark.com/download/?3dmark05.shtml If your system crashes with this test I would say your video card is the culprit.
A few days ago, I put together my first computer. I'm having a problem with my computer shutting itself down when I start to play a video game (World of Warcraft). It happens within a few minutes of starting the game. My guess is that the video card (Gigabyte Geforce 6600 AGP) is drawing too much power from the PSU. Regular websurfing seem to be fine.
Not sure if this makes a difference but, when I was playing around with the AGP settings, turning the AGP speed from 8x to 4x, the computer restarted, instead of shutting down.
When I put my hand over the back of the case closest to the PSU, it is hot. So is the heatsink on the video card. I'm not sure which one is causing the problem. I'm leaning toward the PSU because the entire case/psu was $33. If anyone can assert with a high level of confidence that it is the PSU causing the problem, I'll spring for a new one ASAP. however, I'm not too knowledgable about this issue so I don't want to spring the money right away.
CPU temperature seems to be between 44C and 57C. Motherboard temp is ~38C. Mobo CPU thermal throttle is set at 70C.
Here's my current system
Video Card - Gigabyte Geforce 6600 AGP 256MB, GV-N66256DP
Case - Broadway Com 450W Mid Tower, PSU OKIA P4-450W
Motherboard - ASRock 775V88 Socket T (LGA 775) VIA PT880
CPU - Intel Pentium 4 550 Prescott 800MHz FSB LGA 775
RAM - 1GB DDR400 Generic
HD - Maxtor Diamondmax 10, 160gb
dvd-rom drive, cd-rw drive
OS - Windows 2000 SP2