I have a Pentium 4 1.6gig machine that does not power up, reliably. There is power to the motherboard. I think it may be the main power button, but a side issue with the internet has me uncertain.
The machine will run when it does start just fine.
The only symptoms I can determine that are related, are that if it is shut down normally when the internet is not working then the machine will not start when the power button is pressed.
Sometimes if the power button is cycled many times it will start.
If the internet (Firefox or IE) is operating normally when the machine is shut down, the next time you press the power button always start the machine.