My Xbox will turn on and run games like normal, until it starts having to do more intensive work, then it'll flash a message on screen saying it will turn off due to overheating and then it will flash a single red dot for a few minutes.
I might have it narrowed down to 2 things:
# "Not enough power" Some people say that the fan not spinning up might be because of the power supply. That could be the issue, mine is very old and shares an outlet extender with 2 tv's (flatscreen and CRT) and another console.
# "broken fan" I did take apart my console a few months ago to see why it would run so incredibly hot and sign me out of live. I made especial care not to potentially damage the fan or any cooling component, but accidents do happen.