1999 Pontiac Grand Am 1999 Grand Am blower motor

When the problem occurs, I'd check the circuits at the blower relay. The orange wire is only hot for high-speed, voltage goes to coil side of relay. When relay is energized, that line with the arrow head on it, moves to the high-speed blower fuse. When the relay is not energized there is continuity for the lower speeds, blue wire carries voltage for lower speeds.

You could use jumper wires with inline fuse for voltage and ground to blower motor, blower motor should run as long as the wires are hooked up. If it still stops, I suspects a faulty blower motor.