Originally Posted by
AOD_Guybrush
This may be a possibility - check to make sure your graphics card fan is running (and hasn't burned out). If the fan stops, or if it isn't providing enough heat dissipation, your computer will shut down automatically. If it's far enough gone, you could have burned the card up completely (by overheating).
The reason I think this might be the case - you said it only turns on after you've done x multiple times (or... the computer has cooled off enough to turn back on).
Panda's suggestion is also a likely culprit, but if this has happened multiple times, a short would have done considerably more damage.