If your only problems are running full screen games then I would suspect heat or an inadequate power supply.
How to know? All I can suggest is trial and error. I would run with an open case and even blow a room fan on the board. See if that changes anything.
You can also get a PSU tester for the power supply but again if the issue is only with full screen games I would suspect your graphics card.
Do you have another card you can try?
Is it freezing? Or does Windows continue to work? If the latter:
Check your Event Viewer via Administrative Tools.
Open the System as well as Application tabs and look for red X errors that coincide with your problem. Details here:
http://support.microsoft.com/kb/308427