My 1 year old fell out of the bed onto his back. Could it be broken or sprung?

In this situation you need to take the little guy to the hospital for an x-ray. There really is no way to say that he does or does not have a fracture of the arm otherwise. In fact at this young age it can even sometimes be hard to tell by x-ray if the arm is fractured. This is because the bones of children are still very soft and pliable. He could have something called a "green-stick fracture". It gets the name from how if you take a living "green" tree branch and try to break it, it will bend and splinter instead of just cleanly breaking. Then when you let pressure off the branch it can go back to looking normal. Combine this with the fact that a 1 year old can not talk to you and tell you what is going on, and the need to take the child to be seen at the hospital becomes even more evident. So, yes, I think you should take him!