Likely reason for not being able to power on an older computer?

Your power supply is more than likely the culprit here, it is probably not letting go of the power that is stored in the capacitors.I do not know the exact science behind why this works, but I do know that if you clear the power it will work. To do this you will need to power off the machine, then unplug it, then press the power button a few times to clear the stored power, plug it back in, and then try to power it on. That should solve the problem.

"It's the power supply. If it's loaded too heavily on startup, and if the capacitors are bad, then it won't easily start up - it'll start, but then think there's a problem because it can't keep itself in regulation, so it'll shut down.

Replace the power supply. If you open up the old one you're very likely to find bulging or leaking capacitors.

-Adam"

While I completely agree with everyone who suggests dodgy power supply as the most likely culprit, a possible alternative, with a much cheaper fix, is flat CMOS battery - occasionally this can cause random weirdness on startup. Worth checking before going for more expensive fixes...

What everyone else said, but I also found that IBM 6221/6223 servers had this quirk- when HyperThreading was enabled in the BIOS. Turn hyperthreading off, and we no longer had to pull the power cord in order to get it to reboot unattended.