PC doesn't power on first few times I press power button

When i press the power button on my PC it really powers on for about a half a second. I can see the fans starting moving, some leds lighting, but afterwards it goes dead. It usually takes a few tries for the computer to really start, after which is works just fine.
Any idea what it could be?

I have a 350 W ATX PSU, nVidia nForce2 mother board, AMD Sempron 2600+ processor. (yes it's an older computer which i'm trying to revive).

Thank you for your time.
 
Looks like a dying PSU. I would suggest you swap in a replacement and see if you're facing the same issue.
 
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