God's Overbearing Grace (John 6:44)

There is no doubt that we are saved by the Lord's grace. However, how much grace does God give us to come to Christ? Does he give us enough to counteract the fall? Do we really need grace after Adam's fall? Or are we so fallen and depraved that without the Lord's grace overwhelming us we would never turn to him? How much of a gift is our faith at the end of the day?