This type of problem is almost always related to overheating. As EzioAs suggested, grab a program that reports GPU temps and see how hot the video card is getting.
If overheating is the cause, you can try adding some fans to the case or rearranging the layout/cables to try to get the video card more airflow. If the computer is more than about 6 months old or the room has a lot of dust, get a can of compressed air and blast it into the video card's fans (while the computer is off and the case is open). If you see a big plume of dust appear, that was probably inhibiting the card's cooling.
Another thing you can try in lieu of replacing the card is to underclock it. Yeah, yeah, you don't want to lose performance. But if the choice is between a game which runs slower vs. a game which doesn't run at all vs. spending a couple hundred bucks for a new video card, slower may be the best option.
EzioAs :
First, check the power cables (2x6pin) if they are plugged in correctly into the card. Might be good to resit the card as well in the meantime.
I doubt it's the power cables. I used a 560ti in a client's build and accidentally didn't plug one of the two 6-pins in fully. The card still worked, just the framerates were much lower. Apparently nVidia has circuitry built into the card to automatically throttle down its performance if it's not getting enough power to run at full speed.