When my daughter was five, she fell off the monkey bars in the playground and broke her forearm just above the wrist. At first, the doctor in emergency said that my daughter was fine and didn't need an x-ray. It was only when I insisted that they did the x-ray and found she had broken both bones just above the wrist. Why do you think they didn't believe us?
Treating children for arm breaks is one of the most common tasks for doctors who work in emergency rooms that deal with fractures. The forearm is vulnerable because of the way the arm reaches out to break a fall.
Why the treating doctor didn't believe you isn't something that can be addressed, but emergency personnel take many things into consideration when making their decisions. Of course, they do make mistakes from time to time and that is why it's important for parents to go with their gut instinct.