This is an interesting question and as one who deals with possession and the like I really do not know the theological answer to this guestion. I have had many experience casting demons out of christians (i am not talking the works of the flesh Gal.5). At the same time though I try not to make experience a determining factor in my theology. I do know that Mark 16:17 says to cast out demons!
Mark 7:27 seems to teach that Deliverance from demons is the "childrens bread".
Luke 4:33 seems to allude to a (believer?) being possessed. Many assume (right or wrong) that this man was a believer since he was in the synagogue. But there is no proof either way.
Luke 13:16 shows a "daughter of Abraham" being bound by Satan. This could be a scripture used to say that a Christian can have a demon but being "bound" does not necessarily mean possession. Many scholars teach that this scripture cannot be applied to Christians post-cross since Christ defeated satan and believers have the Holy Spirit. They teach that a Christian is Jesus possessed not demon possessed.
The word possession is interesting since it implies ownership. As Christians we have been "bought with a price". We are now Christ's own!
I do not believe a Christian can be possessed but possibly demonized.
Many teach that a Christian cannot have a demon in his spirit since it has been sealed with the Holy Spirt of Promise Ephesians 1:13. With this I agree.
But these same teachers also teach that the Soul (mind will emotions) and the Flesh are open to demons (claiming that even Jesus cleansed the temple.)
1 Corinthians 6:19-20 tells us that we are to glorify God in our Bodies and in our Spirits which are His.
What communion does light have with darkness or Christ with Belial? 2 Cor 6:14-16.
I have been unable to rationalize and sort all my thoughts on this subject.
In Zechariah Satan went before the very throne of God.
What about Job? God seems to have allowed this demonization but I do not know if I could consider Job's case a possession. It also teaches that believers have a hedge of protection.
What about Eph. 4:27 what does it mean to give "place (topos)" to the devil? Paul is writting to the Christians at Ephesus.
As Christians the attacks from Satan are real and common the question is whether they are internal or external. Ephesians 6:10-18 tells us of the armor of God and all of this armor appears to be for external protection (Paul is writting to believers).
Most people want to cast out flesh which the Bible teaches has to be crucified. If I had to put a percentage on it 90 percent of our problems are fleshly. Maybe the devil did not make you do it.