Gal. 6:1 Dear brothers and sisters, if another Christian is overcome by some sin, you who are godly should gently and humbly help that person back onto the right path. And be careful not to fall into the same temptation yourself.
Yes, it is possible for Christians to be overcome by sin. To overcome means to be overpowered by, to become a victim to. No one is so strong spiritually that they cannot sin. Paul says that when it happens, the rest of the body of Christ has a loving responsibility to help their fellow Christian be restored to a right relationship with Christ. We are very good at messing this up.
We can be good at pointing out the sin we see in others, but very bad at doing it in a loving way. So often we get on our soap boxes and berate a fallen brother or sister and pat ourselves on the back because it wasn't us. We chide others for doing things we would never imagine ourselves doing. We want to fix others without regard for what needs to be fixed in us.
When dealing with another person's sin, there must be a gentle and humble spirit that drives us. We must understand our own tendencies to wander from the right path. We must understand that if the tables were turned and we were in the other person's shoes, we too might have very well had been the one who had sinned. We must understand that it is only by God's grace at work in us that we are able to keep from sinning at all.
We have a loving responsibility to come alongside of each other and deal with our sins, but not in a condemning, self-righteous way. We should always deal with sin in our lives with the goal of restoration to Christ. We must care enough for each other that we don't make matters worse by taking an "holier than thou" attitude. We too are sinners and could very easily fall into the same temptation that caused our brother to sin.
Dealing with sin in others and ourselves should always be first bathed in humble prayer before God. And, when we do sin, we should not resist the efforts at restoration. The true sign of a heart that is right with God is a humble and contrite heart. When sin is exposed we acknowledge it and confess it, whether it is the Spirit of God or man who exposes it. As we wait before God with broken hearts, we become a holy people, a people after His own heart.