Can you just have a broken bone forever a bone that never heals?

Yes, sounds like you might be describing a nonunion fracture. Nonunion and malunion fractures are identified with defective healing: nonunion describes the failure of a fractured bone to heal and mend after an extended period of time; malunion refers to a fracture that has healed in a deformed position, or with shortening or rotation of the limb.

Yes, that can happen. Depending where it is, and what symptoms it causes, it may or may not need to be treated.