The older the dragon the more aggressive they may become towards each other... this can and will happen with any bearded dragons that are put together. Your biggest problems are:
10 gallon tanks are way too small for one single 3 month old dragon. Putting two in a 10 gallon is just ridiculous at that age. (that's not a shot against you just a fact for future readers - you didn't know) You need to upgrade your tank size. Bearded dragons are territorial at times and putting two dragons that age in a small tank with no room to get away with each other is just a recipe for disaster.
No heat gradient. Be extremely careful you have the proper heat gradient. if you have a high end heat bulb on the one side chances are the other side is not getting cool enough for them to properly regulate themselves. High temps can cause your reptiles to be extremely high strung due to overheating and pain and will cause them to flip out with each other. Prolonged exposure to too high temps will slowly cool them.
You absolutlely have to get that dragon with a broken arm to a vet immediately. You can't let it sit there and suffer like ChristyJ said. Please get it help right away.
You should separte each dragon immediately and upgrade their tanks as soon as possible. I don't know how long or big your dragons are at 3 months, but if they are even remotely at the proper size they need to be in something much longer.. if nothing else for the proper heat gradient.
And to answer your other question, it is still possible even with the proper light, space, heat, food, and everything else that a dragon will fight with other dragons. it happens. there is no way to stop it if it's going to happen unless you keep each dragon separate.
We had a pair grow up together, no problems ever as hatchlings. At about 2 months of age (these were two females) i happened to look in their cage one day and they were scrapping like two boxers.. each up on their hind legs going at it with each other.. i was shocked. swiped one out before any damage occured but if i wasn't there we could have had a serious problem.
brandon
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Dragons by Nature