When we were at WDW in 2004, within 24 hours of arrival my DH and DD (3 at the time) both had to be transported to the emergency room with horrible cases of pneumonia. It was aweful. We spent more of our vacation in the hotel room than anywhere else!
I do have to say, Disney's medical care was the best we've ever received!! I called the front desk and a private shuttle arrived to pick them up within 10 minutes! The driver gave DH the forms to fill out on the way to the emergency room and the nurse was waiting for them when they arrived! They were seen by a doctor, had x-rays, then were taken by shuttle to a pharmacy where their prescriptions were waiting for them. A quick jaunt in to pay and pick up and they were on their way back to the hotel.
Total time from start to fisnish was under 2 hours. I have spent more than 2 hours in the er waiting room waiting to be seen here in SO CAL.
Either Disney has incredible customer service, or they don't want you out of the parks too long!
Good Luck, I'll be sending good vibes you way!