Do possessions still happen implies that they used to and no longer do. Either they never did and still don't. Or they did, and still do, because why would it stop if it was existing in the first place.
Demonic possessions never happened and never will. This is because of one, simple fact. There are no such things as demons. In every reported case of so called "demonic possession, there is a logical and scientific answer. Contaminants(chemicals hat alter brain function, in some cases, drugs such as hallucinogens, epilepsy, schizophrenia(as welll as other mental disorders, etc. So you see, substitute logic for superstition, and there is noting to be afraid of by way of demons.
Well, if you believe in demons...then sure. Why should they stop?
Lots of things that were considered 'demons' have been explained, but there are still situations when an exorcism is the only thing that works. Is it because of real demons, or is it just in their head? Who knows. But it works.
Being a person who interacts with nonphysical entities pretty often, I'd say yes, they can possess people. I've had problems with them. And I've found that some of those problems respond to meds, and some of them respond to magic.
If they ever happened, then they are still happening. If they never happened, well, they sure didn't just suddenly start.
In a sense yes, Demon possesion can happen and how they happen there are various qualities. How they did it in biblical proportions, its probably smaller then today, since most of these, would be classified as mental diseases.
There was a sense where you can be possess by multiple demons like the guy name Legion who posses alot pigs and the guy who went crazy, broke chains and big rocks, a thing you can't do as a regular human being.
Apparently you know the movie, the Rite was based on modern exorcism which was based on a book: The Making of a Modern Exorcist [link]
Although you can't blame all demonic possesion, since there is room for falsifications, still demon possesions can happen.
There are different qualities of who is possessed depending on which sect you beleive in.
Catholic: External physical pain caused by Satan; Demonic Possession, in which Satan takes full possession of a person's body without their knowledge or consent: the victim is therefore morally blameless; Diabolical Oppression, in which there is no loss of consciousness or involuntary action, such as in the biblical Book of Job in which Job was tormented by a series of misfortunes in business, family, and health; Diabolic Obsession, which includes sudden attacks of irrationally obsessive thoughts, usually culminating in suicidal ideation, and typically influences dreams; Diabolic infestation, which affects houses, things, or animals; and Diabolic subjugation, in which a person voluntarily submits to Satan.
Middle ages: manifestation of superhuman strength; speaking in tongues or languages that the person cannot know; the revelation of knowledge, distant or hidden, that the victim cannot know; blasphemic rage and an aversion to holy symbols or relics
Other qualities: Has to acknoledge the Lord's authority. Could also be knowing that the demon dualy lives within him.
Or the greatest thing demons can do, is not to let anyone know they exist. I mean the less they exist the less God or spirits exist?
Demonic possession has never happened because there is no such thing. You see, ancient people were ignorant of the causes of neurological disorders like epilepsy, and when they witnessed a person having a seizure, they thought they were being possessed by demons. They were wrong, of course, but the idea of "demonic possession" became integrated into the Abrahamic religions, or Christianity at least. The reason it still seems to happen today in people who are not epileptic (see the Pentecostal church, for example) is because of "suggestion." It's like non-alcoholic beer: a classic prank is to give non-alcoholic beer to someone who has never had beer before, but tell them it's regular beer. In time, they will act as though they are intoxicated, even though they couldn't possibly be. Demonic possession works the same way. When these fire and brimstone preachers tell their flock what pieces of human garbage they are, they act as though they are having a seizure/possessed by demons, even though they definitely aren't.
The thing to take away from this is that human foolishness is almost always a better explanation than the supernatural, demonic possession being just one example.
There are some interesting theories put forth by anthropologists on the subject of demonic possession and exorcism in other cultures, that still practice these rituals. If you want to know more, I'd start here: [link] and here: [link]
I would say it's a mixture of both. You don't need to be of faith to believe in ghosts. I do, and I don't believe in God. Ghosts are one of the mysteries I would love to explain with science, but like in the episode of MLP: FiM where Twilight tried to figure out how Pinkie's tail worked, some things just aren't possible to explain with science and rationale; it just comes down to accepting it works and not sweating the small stuff.
Our person, our existence all depends on our brain. Our brain is millions of electric signals. When we die those signals stop and our brains decay. When we no longer have a brain we are gone. There is no logical way for such a thing as a ghost existing. Ghosts are tricks of the mind played on fear and irrationality.
You may be interested in reading "Paranormality" by Richard Wiseman for a psychological explanation of why people believe in/experience ghosts, faith healings, alien abductions, telekinesis etc.
For any study into the "paranormal" you have to be very critical about the methodology - is the research being conducted in a scientific manner, are there proper controls, does the conclusion follow from the data or are people reading meaning from the data that is not there.