Can a Christian be possessed by a demon?
Although I tend to believe that demon possession was strictly a first century phenomenon, the acts of certain sociopaths or psychopaths are clearly demonic. Note that I stop short with their acts, as I wonder if it is prudent to go claim "possession". Certainly, some (most?) unconscionable deeds can be attributed to the influence of the Devil, but it may be presumptuous to suggest that such is the result of actual possession. While Satan can tempt us, he can no longer deprive us of our free will.
Many would claim that no modern-day demon possessions exist, all having passed away by the end of the first century A.D. The reason is that casting out demons was a matter of spiritual authority, something demonstrated only by Christ, His apostles, and chosen disciples in the first century A.D.
The demon-possessed often exhibited near super-human abilities as we see in the Gospel of Mark:
Mark 5:3b-4: "And no one was able to bind [the demoniac in the tombs] anymore, even with a chain; because he had often been bound with shackles and chains, and the chains had been torn apart by him and the shackles broken in pieces, and no one was strong enough to subdue him."
Further, in the Book of Acts we read how the special gift of exorcism was limited to a very few:
Acts 19:14-16: "Seven sons of one Sceva, a Jewish chief priest, were [practicing exorcisms]. And the evil spirit answered and said to them, 'I recognize Jesus, and I know about Paul, but who are you?' And the man, in whom was the evil spirit, leaped on them and subdued all of them and overpowered them, so that they fled out of that house naked and wounded" (emphasis added).
Here, the demoniac recognized Paul's authority, something given to him by God -- but certainly not to everyone. And, as an apostle, Paul could lay hands on whomever he felt would further the cause of the Gospel. In doing so he imparted special gifts to them: they too might then cast out demons. This powerfully demonstrated godly authority where the people might then be convinced of the veracity of their (apostles', disciples') words.
Since Scripture has been completed and delivered to us today, and no apostles are alive to impart such gifts, there is no longer any reason for God to allow such possessions. Nonetheless, that does not mean that many people are not influenced by the Devil. The perils of human frailty know no bounds.
As for whether a Christian can be so possessed, this is especially unlikely because God protects them, just as He did with the faithful of the first century (Mary Magdalene, etc.), who were cured of such horrific indwellings.
We should understand that "as we walk in the Light" we are "cleansed of all sin and all unrighteousness" (1 Jn. 1:7, 9). God will not allow us to be tempted (and certainly not possessed) beyond that which we able to bear:
1 Corinthians 10:13: "No temptation has overtaken you but such as is common to man; and God is faithful, who will not allow you to be tempted beyond what you are able, but with the temptation will provide the way of escape also, so that you will be able to endure it" (emphasis added).