My son had this happen at 14 months in the same scenario as Bonnie B... His daycare called me saying he was crying hysterically and holding his arm. They told me another child was tugging on a toy against him and when one jerked it away and he fell backwards that's what did it. I still to this day do NOT believe that story. I think it happened when a teacher/aid grabbed him by the arm to separate them. Its so weird that Bonnie has the same story as me. I wonder if it is common at daycares for this to happen, they even knew that it was nursemaid's elbow.
ANYWAY, I took him to the ER and he sobbed uncontrollably while we waited for the X-ray techs. The doctor finally felt sorry for him since we were waiting so long and said "Just let me try one thing" and popped it back into place. My son was so exhausted from the crying that 60 seconds later (the pain immediately stops when corrected) he was sound asleep.
The doctor said this can also occur from swinging your child by the arms.....lots of people do this. Let me strongly advise against this. You do not want to be the cause of this. I would have been devastated if I had caused my child that much pain for something so easily prevented. The doctor also said to be extra careful with the arm (no swinging, pulling them, or picking them up by the arms) until the age 4. If they make it to 4, its unlikely it will occur again. Luckily we have almost made it to 5 without it happening again.