Does the motor have an electric brake in it? If so, the brake could be dragging or not releasing at all.
Many Z axes (on mills) and X axes (on slant-bed lathes) use brakes to keep the slide from drifting down when you turn the power off. When the power is on, the brake is usually released electrically. Sometimes the brake is separate from the motor, but most of the time it's built right in to the motor to save cost.
The current going to a servo motor is almost directly proportional to the load on the motor. Even when the motor is not moving, current is required to just hold the motor in position. Any servo must "hunt" a little to maintain position, so a motor that is fighting a stuck brake is drawing excess current even at rest.
Other possible causes are bad lubrication, bad ballscrew or thrust bearing, counterweight not working, etc. My bet is on the brake.