Though the battery was low, then just would not start?

My car started having trouble starting for about a week, as though the battery was low, then just would not start. Took the battery to Autozone to have it tested and they said it was fine. When I try and start the car it does not turn over and there is no clicking sound. I made sure that the connections on the battery were really on there, but still nothing. Also, the alternator was replaced about a year ago. Could it be the starter or is there something else that may be the problem?