The first thing that I would be inclined to check if the PC turns itself off when playing games and tends to run fine when surfing the web and using Word etc is overheating.
I would check the fans are all still running are that the PC is free of dust inside.
It could be the power supply but you don't mention anything about your PC. Need to know speed of processor, operating system and amount of memory, also is it new or an older PC, if older did it used to work ok and has only started doing this?
Bit more info mate. Don't see what the refresh rate would have to do with it. If you set that too high you would just get either nothing on the screen or garbage as the monitor couldn't display it, it wouldn't just fail part way through playing a game.
It sounds to me like the graphics card is overheating, check the fans first.