It can happen. I have had a video card shoot fire before.
Addition: The card failed with no overclocking at the time of failure. It just shut off over night and then in the morning when I found it, I opened the case to investigate. Reset the power supply, powered on, and flames shot off the video card. BFG did not question me when they went through their troubleshooting process for RMA. I was like, it shot fire out of a component on it. Haha.
Also, I have seen power supplies get very hot and burn up. Like people are saying, don't cheap out on a power supply. All the other times were technition mistakes back in the day I saw turn badly for them from connecting wires incorrectly or things not seated properly. You know something went wrong when you hear someone turn the system on and then a string of expletives following.
But to answer your question more directly, you will not normaly see computers just catch fire. Components are mostly made efficiently and safely today with tolerance and failure limits to prevent further damage and user safety. There is always exceptions, but in most cases, the component that dies just dies and computer will either give errors or just shut off.