Yes, a belt or belt tensioner can cause this kind of problem. With the engine off, you might try placing a socket on the alternator pulley nut and attempting to rotate the pulley by hand.
This should be very difficult to do. If the pulley rotates somewhat freely then the belt is the fault more than likely.
Another more obscure cause could be a worn brush or weak brush spring which is allowing the brush to hop off of the commutator on the alternator armature.