What causes HVAC blower motors to go bad?
HVAC blower motors, which are almost always of the INDUCTION motor type, fail for a variety of reasons.  I'll list what are common reasons:

A winding failure, usually an open.  These can be caused by a manufacturing defect or just as a result of thermal cycling.  Shorts between windings can also occur often due to conductors rubbing on each other or against the pole material.
Bearings can fail often due to foreign material that collects.
The rotor laminations can rust and cause binding.

As noted in the comments, these often have a starter circuit with a capacitor which can fail.  This generally leaves the motor operational but it cannot be started.
Usually the first thing to check is the starter circuit and capacitor.