Can Christians be demon-possessed?

No, Christians cannot be demon-possessed. Possession implies ownership, and Christians are not owned by the devil. Instead, we have been bought with a price, the blood of Jesus Christ (1 Cor. 6:20; 7:23). Also, a demon cannot dwell in the house of the Lord because of the surpassing greatness of God's glory and purity. As Christians, we are the house of God (Heb. 10:21; 1 Tim. 3:15) when we are indwelt by God when we receive Christ (John 14:23).

Furthermore, the Bible says "Greater is he that is in you than he that is in the world," (1 John 4:4). So, it would make no sense to affirm that demonic presence could actually inhabit a person who is also indwelt by God.

Furthermore, Jesus told a parable about casting demons out of someone that can shed some light here:

Matt. 12:43-45, “Now when the unclean spirit goes out of a man, it passes through waterless places, seeking rest, and does not find it. 44 “Then it says, ‘I will return to my house from which I came’; and when it comes, it finds it unoccupied, swept, and put in order. 45 “Then it goes, and takes along with it seven other spirits more wicked than itself, and they go in and live there; and the last state of that man becomes worse than the first. That is the way it will also be with this evil generation.”

As you can see, when someone is possessed and the demon is then cast out, the unclean spirit tries to come back later. Notice that Jesus says that in this case the house is found unoccupied. This most probably means that a person is not a Christian since the house is empty; that is, he does not have the indwelling presence of God. An empty house allows other spirits to enter into this person and his condition is worse than it was before.

So, it seems clear in scripture that a Christian cannot be demon-possessed.