Interestingly I had a similar experience three years ago as well. I would cough non-stop and no medication was of any help. It lasted three months and then slowly disappeared. The weeks after I still had some respiratory problems (whistling when breathing), but nothing too annoying compared to the cough. Without joking, this was far from being an agreeable period and it impacted my studies quite a lot as coughing all day made me very tired and I was unable to study properly. Doctors never found what had happened.