While I don’t believe that it is possible for a genuine believer to be possessed, there can be various levels of oppression. These can take the form of an increased intensity of temptation, or of strong disordered thoughts, feelings or impulses that seem too powerful to resist. A person experiencing this could be called troubled or oppressed by demons, or demonized.
When talking about the demonic affliction of believers, I favor a descriptive approach rather than one that relies on terminology that can be misunderstood or argued over. Rather than using a certain word like possessed or demonized, I prefer to talk about a person as having problems or troubles that appear to have a demonic component.
Since there is no objective ‘lab test’ for the presence of demons, we are forced to interpret what we see or hear of sense, and by what we believe to be the voice of the Holy Spirit. I will give two examples of prominent Christian leaders who changed their minds on this issue.
Merrill Unger earned his doctorate at Dallas Seminary and later taught there. His doctoral thesis was published in 1952 under the title Biblical Demonology. This work became the standard in its field for decades among conservative evangelicals. Unger taught in it that a Christian cannot be invaded by demons. Over the years he received many letters from missionaries and pastors who described cases of people who were known to be sincere believers, and who were clearly oppressed and almost controlled by clear demonic manifestations. Unger traveled widely, and over the years witnessed many cases of demonic manifestations and the expulsion of demons in people who then gave glory to God, showing evidence salvation and the joy of the Lord. Unger reversed his position, publishing What Demons Can Do To Saints in 1972.
Charles Swindoll studied at Dallas Seminary while Unger taught there. He also grew up with the standard teaching of Unger’s earlier book. After decades in the pastorate, he also reversed his position, saying as Unger later did, that in cases of persistent sin, occult involvement or deep lasting unforgiveness, that demonic oppression of believers was, as far as he was concerned, a reality. In his book, Demonism, he said, “I have worked personally with troubled, anguished Christians for many years. On a few occasions I have assisted in a painful process of relieving them of demons.”
Both Unger and Swindoll stressed that they were not saying that these believers were completely possessed in the same manner as an unbeliever could be, but that true believers could be oppressed to the point where they needed deliverance in order to follow Christ as they used to. Your questions are always welcomed.