1992 Acura Integra Battery Not Getting Power

My battery light started coming on while driving and it got to the point that my car completely died out. I figured my alternator wasn't feeding the battery and I would need a new alternator. My belt was squeeling prior to my car not running. I had my alternator belt tightened and then the problem was solved, the battery light wasn't coming up and my alternator was charging my battery properly again. Everything was fine, now just a week later, today for the first time the battery light is coming up again. I havent had a chance to check the belt or anything, drove straight home before car died out like last time. When I tightened the belt, I tested the battery and the alternator were both reading good. Do I need a new alternator? If so how come it worked for a week no problem, back to normal by just tightening the belt?

I appreciate good advice, thanks.