I had a computer behave exactly as you describe. It was quite some time ago, I don't remember the specs. Anyway, I also thought it was the video card causing the problem, but it wasn't. The motherboard ended up being the failed piece of hardware in my case.
Does anybody know where I can buy just the 4 pin fan connectors to make my own cables the length that I need them to be? I could only find 8.5" extensions.