Odd problem on a relative's comp

My grandmother’s computer is behaving rather strangely. It starts up fine, but if you wait a little bit and try to run anything, it suddenly shuts off. It is running windows 98, so I would have expected a bsod rather than a sudden shutdown. After it shuts off like this it can only be turned back on by unplugging it and then plugging it back in after a few minutes.

To me this screams bad power supply, but several relatives claim that it just had a new one put in not more than a year or 2 ago. Anyone have any clue?