In Biblical times, being possessed by demons was a way to describe any sort of mental or physical condition that no one could explain. Today, we have medical and pyschological knowledge that the ancient world didn't have, so we don't have to blame it on the devil. Epileptic seizures, bipolar disorders, depression, multiple personalities...all these and many more conditions which are explainable and treatable today would have been blamed on the devil in the ancient world. But that's also the world in which a woman's head needed to be covered and she couldn't speak in church....
However, there are a very few documented cases of real demonic possession. The Catholic Church is very, very slow to admit to demonic possession, because we understand that most cases are really cases of schizophrenia or mental unbalance, that is treated with intensive therapy and drug therapy. It is the common held belief that every diocese in the Catholic Church has an exorcist-a very holy priest whose job it is to perform the rituals of exorcism in the rare case of documented demonic possession.