Your problem will most likely be 1) the case on/off switch bad, or 2) the PSU has quit working properly.
For #1), try swapping the case reset switch for the case on/off switch...they do the same thing...so, if the power supply is ok, the "new" switch should function just fine, and you can turn the machine on by hitting the reset switch. (you can also remove the connectors from the on/off switch where they connect to the pins on the motherboard and short those two pins with a small screwdriver or something else metal for a test. Be sure to fully disconnect the computer power cord from any power source first).
If this doesn't work, you will need some way to test the PSU with a volt meter.
I doubt your problem is a bad motherboard or CPU...it COULD be a bad, or not plugged in good...stick of RAM. Bad motherboards or CPUs do not work awhile and not work for a time...go and come etc. They are either good or bad...not somewhere in between.