Will not start

Yesterday my car did not want to start. The lights would not turn on and the remote key would not respond and unlock the car either. So I thought it was a dead battery. I took it in and it was healthy. Next I checked the alternator and took it in to be checked. Turned out the alternator was bad. After that it started to run! It ran for a whole day then the next day it didn't unlock with the remote key again so I was confused. I called triple A and they came to unlock the car and tested the battery and it was fine. But now the car won't start. Any suggestions or ideas as to what is going on here? Any help is welcome. Thanks!