You might have seen a number of reports on "Health Tourists" in the media recently. Some of the figures banded about are based on a flawed study from Newham. I've seen a couple of these "tourists", the most recent was a woman from India who landed in London, went to her friends in Newham and a day later was calling an ambulance (my ambulance if you haven't guessed) as she was in labour. I did tell her that she wasn't supposed to fly after the 35th week of pregnancy, but she didn't speak English.
I've also dealt with a number of "Granny came off the flight and got chest pain" cases. I remember one such case that her test results showed that she'd had her heart attack a day before flying.
My personal experience is that it isn't actually that common, and I suspect the media like this story as it is in the same league of "news" as reporting on illegal immigrants.