Might it happen that a bone be broken without it being in "constant pain"?

Yes: The brain sometimes turns off the pain to concentrate on other things, people have a variable amount of pain tolerances. There was a martial art patient who had such high pain tolerance that he ignored the appendix that had ruptured and died by the time he came to the hospital. On and off pain is worthy of an evaluation.

Yes: Constant pain is not always seen with a broken bone.