I think its an electrical problem? 93 cavalier

I have 1993 Cavalier. The past few weeks its has had trouble starting a couple times. Yesterday it wouldnt start at all so I went to Walmart and had a new battery put in and it started just fine. Now today I went to start it and it wouldnt start again so I had a mechanic check things out, he tested the alternator and battery and said they were both fine. The car will start if I jump it or charge the battery for a bit first so my question is ,what does that mean? Where do I start? I guess its like something is draining the battery while the car is off. Ive made sure all the lights are off and I dont have any after market tv or sterios or anything like that installed. Any help is appreciated