Don't know anymore

I just purchased a 1993 Mercury Sable 3.8. I'm going to try my best to describe any problems (my husband is the one who knows the right names). The belt on the alternator was squeaking badly. My husband tightened the pulleys and he realized later that they became loose again. He also realized that the battery light was coming on (this is a new battery). When he shut off the lights, radio, and heater the battery light would go off. Somehow he realized that when the car was running and he took one of the battery cables off, the squeaking would stop. All this made him think that we need a new alternator. Now first would you agree with that? Anyway when he went to start the car up it wouldn't start. We heard clicking from the computer but know thats its not that (tested it on our other sable and it worked). Could the car not be starting because of the alternator?