1996 cavalier starting problems

I have a weird problem I hope someone can help me with. The car does not start all the time. You turn it to the on position and you can hear the electrical system turn on. You turn the key to the start postion...some times it starts and sometimes it doesn't. It does not make any sound like a clicking sound when your batterys dead...there is no noise. It has done it both when the car is hot and cold. Somone suggested the key maybe worn down so I went and had a new key made from the key number. It does this with all 4 keys to the car. If anyone has had this problem or have a solution please respond. 2.2l engine auto transmission.

Thanks
Chad