If the start button is pressed some times it will light, or flash 4 or 5 times, also the Mute button to the right of the start button will also light up some times.
I suspect the flashing leds are POST error codes.
To find out what they mean, you will need to go to HP's web site, and look in the support section for your model. There is doc there for the codes, and what components are likely the problem.
I would also check into the power you have for that system. Flacky power can cause all sorts of problems.
I looked for Model C500, it listed over twenty models. Have a look for a model sticker, often on the bottom of the system to identify the exact model. Error codes may ( probably ) differ on some models, depending on BIOS level and BIOS maker.