You get sick from bacteria or viruses, not rain. But being in situations where you get wet and wind up chilled, which drops your body temperature, can make you more susceptible to the effects of bacteria or viruses.
My mom used to say that whenever I was playing in the rain. I don't remember getting sick each and every time I played in the rain. If however, your immunity is low, the wet conditions may cause you to become sick.
It is an old wives tale that you get sick from playing in the rain. Illnesses are caused by either a virus or a bacteria and not from getting cold and wet, although getting cold and wet will lower the bodies resistance against viruses and bacteria, but that is all.
I used to play a lot, always drifting wet under the rain. Luckily, I can't remember getting sick. It's even one of the most memorable spontaneous activities I had as a kid. Though, I wont recommend it to those who have low body resistance. It holds you susceptible to illness-causing diseases.