A genuine believer cannot be possessed by demons because he has the Holy Spirit living in him. Light cannot fellowship with darkness, and so when the Holy Spirit comes in, demons are shut out. The devil can send a lying spirit to a believer to trick him into believing that he is possessed by a demon when, in fact, it can be an oppression.
Because unbelievers are in the world, and the devil is the god of this world, they are in the complete control of demons. If they get involved in the occult or pagan false gods or even false "pseudo-christian" religions, they can be possessed with demons because there is no Holy Spirit in them to keep them out. These people can look very normal and live normal lives. The demons don't need to manifest because they have complete control anyway. Demons don't like to advertise their presence, and this is why many (not all) mental patients are demon possessed (but it takes careful counselling to distinguish between the two, and fools can rush in where angels fear to tread, causing damage to people's lives and withholding good medical treatment where it is necessary and helpful).
When a person receives Christ, if he is demon possessed, the demon may manifest and it can be cast out to enable that person to be filled with the Spirit. When a person receives Christ he is taken out of the kingdom of darkness and put into the kingdom of God, so the demon has no further right to possess that person so it gets kicked out. But it can still control that person through oppression and obsession.
This is what happened to me. I received Christ and because of some occult practices in my parental home, there was an influence there. When I became a Christian, the obsessive demon transformed itself into a religious spirit of pride and piousness. Two years later, I was ministered to, and it was broken off me. Because I wasn't possessed, nothing came out screaming or anything like that, but I had a vision of Jesus and I ended up on the floor weeping my eyes out. When I got up, I was freed and changed. After that, I had the wonderful experience of God revealing Himself to me on a personal level.
What this tells me is that believers can be influenced by obsessive and oppressive demons. The gift of discerning of the spirits can reveal this to a wise and senstive counsellor. I don't believe in dealing with demons in public. I believe that ministry and deliverance belong in the counselling room. Counselling can take place through a check-list of possible involvements in the occult, immoral relationships, pagan and pseudo-christian cults, traumatic events, etc. As the checklist is gone through and involvements, bitterness, fears, etc., are renounced, a demon may manifest itself, and therefore can be ejected at that point. Often, when a person goes through the checklist, the only manifestation is a real sense of freedom and rejoicing.
What happens when a person is possessed, or if a believer remains obsessed or oppressed by demons, is that they have done something that has invited the demon in, and like rats that cannot be elimated until you get rid of the garbage they feed on, demons feel that they the right to be there because they were invited in because the garbage is there for them to feed on. So, good counselling gets rid of the garbage, and then the demon has nothing to feed on, and so has no further right to be there. What makes it difficult to get some free from demon influence is that the person trying to minister does not get rid of the garbage first. That is why we see some dramatic events when people try casting demons out of people in public altar calls without the background counselling to identify the garbage and to get rid of it first. But once the garbage is gone, the demon will often say, "There's nothing here for me any longer so I'll have to go." And then deliverance will come a lot easier.