hmm blue screen after playing games for 15 minutes there could a few things that may be wrong.
1. Driver/hardware conflicts
2. Over Heating, but I doubt it usually computers just shut off or restart (or just explode) when they over heat but I've seen a few BSOD's when cards over heat, so wouldn't hurt to look and see if your case is well vented.
3. low memory, again doesn't seem to fit the bill since I've got 512MB of ram and I can play doom 3 with no issues ( ATI 9800 PRO AIW), make sure your not running 3 trillion apps in the background.
4. Software Conflicts, OS Conflicts with hardware, act of [insert god].
5.bad hardware (how long have you had your 6800?)
6. The universe just gave you the finger.
Make sure your OS is up to date, Make sure your graphic drivers are up to date, make sure the game(s) is up to date, make sure you have no conflicting drivers/hardware (device manager) and make sure your memory pool isn't filled to the brim with background apps.