I'm thinking either the CPU or Video card (if any) is overheating... She says after that waiting a half hour, it gets farther into the boot process than it does when immediately starting the computer back up.... My impression is that the cpu/gpu has had time to cool a little and that's why it's getting farther, after waiting..
PSU would be my next guess, however it's much easier to try another PSU than it is to add another layer of thermal grease to the cpu/gpu, so it would be better to try another PSU first, of equal or greater wattage..