Our Hamilton Beach food mixer Model 210 [1960...

These motors can fail in a number of ways:
Open windings - this may result in a bad spot, a totally dead motor, lack of power, or excessive sparking. Windings can open from a major overload/overheating episode either melting the wire or solder connections (the latter on the armature usually), defective manufacturing and thermal cycling, or damage during servicing.
However, the source of the open may not be the windings but a blown thermal fuse - see below.
Shorted windings - this may result in excessive current, severe sparking, reduced speed and power, and overheating. The thermal protector, fuse, or circuit breaker may trip. Continuing to run such a motor may result in a meltdown or burned coils and insulation - i.e., a burned out motor.
Worn carbon brushes - while these usually last for the life of the appliance, this is not always the case. The result could be erratic or sluggish operation, excessive sparking, damage to the commutator, or a motor that does a pretty good imitation of a paper weight :-).