First thought would be to check that the fan isn`t clogged with dust or debris. If it`s sticking it will cause a cut out and you will get what you describe - no power up. Sounds like it does this sometimes with yours. Next replace the fan maybe. It could be failing or the bearing is sticking because of lack of lubrication. (Once it starts spinning the lube gets redistributed?)
The only other cause might be cured by removing the battery and power supply. Hold the power on button for 30 seconds and then reboot using the power supply. This discharges the capacitor on the power board. You can then reboot and put the battery back in.