instant shutdown and no restart

The last couple of weeks my computer has randomly shutdown several of times for no apparent reason. A couple of times when I was using it and other times when it was in sleep mode.
It is like the power is removed but I still see led light on the start button. When I try to restart by pushing the button nothing happens. I have to unplug the power supply cord for 15sec or so. Then I use the start button and all boots as normal.
I have thought of the power supply but that doesn't explain the led light on.
Is there a way to test the power supply?
Or is there something else that can cause a shutdown like this. CPU overheating?
Older Dell machine:
1.6gig hz cpu
512meg ram
128meg video
120 and 180gig hd's
WinXP Pro (all the updates)

 

It could be overheating, it could be a bad power supply, it could be capacitors on the motheboard are failing. Unplug the power cord, open the case and have a look inside. Especially take a look at the fans and heatsinks. If they're full of dust and lint and crud, then overheating is a definite possibility. Give the fans a poke. Do they spin easily and freely? Is there a bit of "bounce back" when it stops spinning? If you said yes to both the fan questions, and your heatsinks aren't full of crud then it probably isn't overheating. Look at the capacitors. They're the small cylinders on the main board and they're about 1/4" to 3/4" tall and less than 1/2" in diameter. They should all be PERFECTLY flat on the top (usually silver with a cross). If they are 'popped' even slightly, and/or look like something is leaking out (usually comes from the middle of the cross) then your motherboard is toast. Sometimes they'll be flat but there will be little black or dark brown specks stuck to the top right in the middle; it may look like dust or debris. It isn't. It's a leaking capacitor and it means your motherboard is toast. If all these "tests" pass, then there's a very good chance it's the power supply. But it may also be a bad video card, memory stick, etc.... but these are the main things to look for in step 1.