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.