Usually when the blower does not work on high it is a faulty blower switch. Though it is merely a guess, the better thing to do would be to inspect the power coming out of the switch and the blower resistor to see what the exact reason is. Somewhere there is a loss of voltage causing your problem. High speed sometimes has it's own circuit and protection. Feel free to stop by for assistance if needed.
-Curt's Service Team 248-545-0500
turn your ignition to on!, turn your fan switch to high!, under your dash by the blower motor you should find a 7 wire plug, wiggle it, if your fan starts, replace that plug, I have replaced that plug in both of my chevy trucks.