I, too, know people who teach that true believers can be possessed by devils (usually called demon-possession). This includes some men that I think a great deal of personally. However, in this I disagree with them. I have left many of the teachings that I had in my youth. However, I find that the early teaching I received on this still holds. The question was asked, How can one person be indwelled by the Holy Spirit of God and be possessed by a devil at the same time? I have still never heard this question adequately answered.
I have also seen that most who teach this doctrine tend to rely on personal experiences. Someone that they knew was saved was also obviously possessed. However, this assumes two things. One, that they can know with assurance who is saved; and two, that they can know with assurance who is possessed. Both of these are problematic to me. We do not live in the apostolic age of signs and miracles. How do we have this certain knowledge? I have seen too many people who claimed to to have special information from God and I have sometimes seen that information to be wrong. Since God does not make mistakes, that proves their knowledge to be faulty.
Now, although I know of no particular verse that directly states that devils cannot possess believers, there are many verses that demonstrate the impossibility of mixing opposites. Here are a few of these verses:
Matthew 6:24 No man can serve two masters: for either he will hate the one, and love the other; or else he will hold to the one, and despise the other. Ye cannot serve God and mammon.
1 Corinthians 10:21 Ye cannot drink the cup of the Lord, and the cup of devils: ye cannot be partakers of the Lord's table, and of the table of devils.
2 Corinthians 6:15 And what concord hath Christ with Belial? or what part hath he that believeth with an infidel?
James 3:11 Doth a fountain send forth at the same place sweet water and bitter?
Notice especially 2 Corinthians 6:15. There is no concord between Christ and Belial. There can be no agreement between Christ and the devil. How can someone be a member of the body of Christ (Ephesians 5:30) and be possessed by a devil? There is an old proverb that says you cannot mix oil and water. These verses show that you cannot mix God and devils. A person who is "in Christ, he is a new creature: old things are passed away; behold, all things are become new" (2 Corinthians 5:17). We know he may fall into sin and even rebel against God. However, if he can also be possessed by devils, he seemingly has no protection as a Christian. I just do not believe it is possible. The believer will have spiritual battles; he may even be (as the Bible teachers used to teach) oppressed by Satan or by devils. However, to think of him as being possessed by devils while being possessed by the Holy Ghost is simply a contradiction in terms. I hope this helps.