No CMOS?
If not I'd look for bad caps like gunnar posted. Usually the signs are your PC freezes/locks up intermittently. The lockups get more frequent until one day it just won't turn on any more.
Here is a link to show you what to look for.
Badcaps.net - Badcaps Home
Basically, look at the larger capacitors for any swelling on top. They should be perfectly flat. If there is any swelling on top it's bad or going bad.