From your description it sounds like something interfering with the fan, like a matted lump of dust and fluff rubbing on the fan vanes. If the fan is actually prevented from spinning at a reasonable speed the firmware will probably detect this and could well decline to start the computer for fear of the processor overheating.
First of all, try poking a pin through the air vents to see if you can dislodge any matted dust then try and suck it out with a vacuum cleaner. However, you will probably need to disassemble the laptop (not that hard) in order to gain full access to the fan.