Crashing while testing each module individually is strange because previously it would actually complete the bootup and run the games for while, correct?
I guess now the next step will be to replicate the problem.
You said that it happens most frequently on game A? What's the difference between the games? If it is a power supply issue, my next question would be why would game A cause it to happen and not game C? And why does a mem test cause it to happen (which still makes me think it is a memory issue some how).
Oh, and is your system overclocked? Because, that is something else that can cause this behavior. If it is, you may have to adjust the frequency and/or voltage. But you better know what you are doing if you are adjusting these, because you can fry your system -- literally.