Did you check like Joe said to see if it the alternator's output is 12.8 or above at idle ? A regular car battery will sit right around 12.8 (sometimes even with a bad cell). Anything above 12.8 will be a charging situation but obviously, if there's a bad cell, you're going to have troubles when the battery's under load (like when it's cranking to start).
If the alternator's charging in the proper range, as Joe suggested, it's not the alternator that's the problem. Highly unlikely that it's the belt either-if it was, you'd definitely hear the belt screeching under load and if you tightened the snot out of the belt, like Rocketman said, you could prematurely wear out the water pump bearings. The belt should be tight but not crazy-tight.
What we need to figure out is if the alternator is really charging when the light is on on the dashboard. Is the light on only for a few seconds every now and then or is it on constantly for long periods of time ?
A car can run a LONG time on just battery power with no charging alternator.
If the light is only on for a few seconds at a time then I'd say your no-start issues aren't because of an alternator / charging problem.
Do you have any more details on this problem ?
Dave