No, this is not what the Bible says. One of the Bible's greatest truths is that we are saved only by God's grace, and never by how good we are. Our only hope is in Jesus Christ, because by His death on the cross He completely paid the penalty for all our sins -- past, present and future. The Bible says, "He forgave us all our sins... he took it away, nailing it to the cross" (Colossians 2:13-14).
That should give you great comfort, because what you say is true: No matter who we are, we're never completely free from sin. Nor will we ever be, until God takes us home to heaven. The Apostle Paul was the greatest Christian who ever lived -- and yet even he wasn't perfect or free from sin, and he knew he was saved only by God's grace. He declared, "Not that I... have already been made perfect, but I press on to take hold of that for which Christ Jesus took hold of me" (Philippians 3:12).
Does this mean we shouldn't be concerned when we sin, or we shouldn't try -- with God's help -- to be better? No, of course not. God's will is for us to become more and more like Christ. Make that your goal, and then learn to walk with Christ every day as you pray, read His Word and have fellowship with other believers. The Bible says, "Let the word of Christ dwell in you richly... with gratitude in your hearts to God" (Colossians 3:16).