It sounds to me like you power supply can not supply enough current for two motors.
Or his motor driver is not able to handle the load and is starting to act up. Perhaps one of the motor uses up more current do to some mechanic flaw in his robot/car and the motor driver is not able to handle it. He said the motor driver is buzzing, not the motors. Then again it could be the power supply. Who knows when we have no specs.
@ timlee we need to know what motor driver, motor, and power supply your using.
Best of luck!