Both Miner and Jim are right on. This is very application and implementation specific.
First of all what type of motor is this? AC or DC, brushless or brushed, direct drive or some type of transmission (pulley, gears, chain, etc), permanent magnet or wound field, induction or armature, (It goes on and on...)
How big is this motor? smaller than a walnut, as big as a bus, somewhere in between?
There are some failure modes that are pretty common though. Miner mentioned one of them which is bearing failure. In general over-heating is a common failure in many motor designs due to misapplication (wrong load, wrong duty cycle, improper cooling). Depending on the motor type there could be issues with controls for position, commutation, speed, and more. Poor manufacturing can show up as short circuits in the wiring or decreased power output. There can be balance issues with shafts which lead to vibration issues and the problems associated with that. There can be vibration in general from stators or external sources. I could go on but you should be starting to get the idea.
It really does come down to your application and the motor design. Your best source of information is a motor manufacturer or find a motor consultant to help you. Some of the major motor manufacturers (and some of the smaller ones as well) have put together publications on motor design as well as motor troubleshooting guides. This may be a place to start.
You may want to avoid being too specific in your quality requirements and lean more toward a performance based requirement. These would be the type of items that could be validated through testing in the application. This prevents you from limiting the motor designer in the methods that they can use to provide a motor that meets your needs.
If it is not already apparent I do work for a small (fraction horsepower) motor manufacturer.