desktop won't power up when off sleep works

If the computer is shutdown normally it will not start. Nothing happens when the power button is pressed. If the computer was put to sleep it will recover. This works for many days until it is unintentionally shut down.

If it is left alone for a long time, sometimes days, the power button will blink and it can be started. If it works why can it not start when off. Is there something in the bios, like an anti theft, that will keep it from starting?

Thanks for any help.