Why does my computer power off suddenly and without warning?

Got another power supply you can swap in for testing? If not debris in the case, could be a power supply failing.

I think overheating is very likely, if as you say, it shuts down then doesn't turn on straight away(it could be cooling down). I have heard of that as a symptom or possible symptom of a system overheating and may have encountered it myself once or twice. Typical candidate for overheating would be CPU. Run speedfan, and see if you see a Temp go up and up and reach a certain point, then the computers shuts off!! or touch the heatsink and see if it is too hot to touch.. ! (the inside of a computer won't kill you.. it's DC and not high voltage. Though many might turn a computer off before doing so.. it's fine.. i've done it!

You may want to take precautions for static. Some people worry about that.

If it is a old computer it might have lots of dust and cak inside the case, causing it to overheat. Take the case off and have a look, then give it a clean if need be.

If it's not heat, it may be a bad heat sensor. See if there's any log entries about why the box shuts down.

(I'd still go with replacing the PSU first though.)