Older computer have always crashed in games

Hello experts

Six years ago I made my homebuilt system. This was the first (and only) time I have done so - and I didn't have anyone helping me. At first glance it looked as though I succeeded, but it wasn't long before I had my first in-game crash. Then again, and again, and again...
I used all my money on the build and didn't have any left for a professional to diagnose it (and I probably still don't). So after trying everything I could think of with my limited knowledge (and Google) I just learned to live with it. But now, 6 years later, I want to try again. I don't have any more knowledge about computers than I had then - so I'm asking you to please help me.

The problem:
The crashes ONLY happen in 3d applications (i.e. Crysis, Driver: San Fransisco, OMSI, - you name it!). I'm "allowed" to play for a various amount of time (sometimes it crashes within seconds, other times I can play for up to an hour - but generally ten to fifteen minutes). I hope there is no need to tell that I of course have made several clean installs of Windows without luck. Other than the crashes the computer lives up to all my expectations, and it performs very well (for a computer this old).

The system:
Mainboard: ALiveSATA2-GLAN
Processor: AMD Athlon 64 X2 5200+ (2C 2.6GHz IMC, 2x1MB L2)
Memory: 2x OCZ2G8001GB DIMM DDR2 PC2-6400U DDR2-800
Graphics processor: NVIDIA GeForce 8800 GTX (768MB DDR3 1.8GHz 384-bit)
Hard disk: Seagate ST3160211AS (160GB, SATA150, 2MB Cache)
Optical drive: Optiarc DVD RW AD-5170A
Monitor: SAMSUNG LE46A-866 (TV), previously: Mirai 19" and one other of which I don't recall the name
Power supply unit: MS-TECH MS-N650-VAL

At one point I suspected the PSU for not delivering enough power. Therefore I've tested the system with dual PSU (the second PSU was a Colors IT 400W which came with the cabinet). But it made no difference.

Operating system: Windows 7 Ultimate (service pack 1), previously: Windows XP, Windows Vista

I guess there's nothing else to say than I really hope someone is able to pinpoint what's wrong (and
has been wrong for all of it's "living" time   Thanks in advance.

PS: If you need ANY other information, just ask.