Of course if it's too low it won't work, but your typical "low" refrigerant usually causes the A/C to get too cold and the evaporator will ice over and block the airflow. At least that's how it happens in a home A/C where the thermostat is inside away from the condenser and its job is to turn on the compressor when the temp is above the set point. On an auto A/C system, the thermostat is right at/on the evaporator and its job is to keep the evaporator from getting to 32 degrees and icing up, the temperature of the cabin is adjusted by how much of the ambient air bypasses the compressor.
If the refrigerant were low, the condenser would be on the verge of icing up almost constantly after the first time the compressor kicked on and brought the condenser down to the low 30's. The compressor would probably cycle oddly, but I would expect the air to continue to be chilled.
I'm no HVAC tech by any means, but if they just dumped refrigerant in without actually testing the pressure on both sides of the system, they're not following procedure.
Brent