Can Christians be demon possessed?
No. But a demon's goal is to disturb spreading the truth of the gospel. Satan is not as powerful as God; he is not omniscient (all-knowing) or omnipresent (able to be everywhere at once). So, naturally, he focuses with zeal mostly on those who are serving God.
"For though we walk in the flesh, we do not war according to the flesh, for the weapons of our warfare are not of the flesh, but divinely powerful for the destruction of fortresses. We are destroying speculations and every lofty thing raised up against the knowledge of God, and we are taking every thought captive to the obedience of Christ, and we are ready to punish all disobedience, whenever your obedience is complete" (2 Corinthians 10.3-6; NASB).
All throughout the Bible we find examples of times when demons inflicted physical pain or illness upon people. (Read Acts 8 for an example of this.) Demons can reside in a person and have direct control over him. (See Mark 5.2-4; 8; 17.15; Luke 8.27-31.)
The question is whether Christians can be inhabited by demons. There is no Scripture reference that says those who are saved, true Bible believing Christians--those indwelt by the Holy Spirit--can also be possessed by a demon. However, believers are vulnerable to the effects of demonic activity.
James 4.7 says: "Submit yourselves, then, to God. Resist the devil and he will flee from you."
It is clear that if we do not take the action of resisting the devil--or demons--then they are in our presence, working havoc in our lives. But Christians have the power of Almighty God to call upon to aid them in resisting the devil.