A not uncommon cause of severe pain in the ribs, when X-ray rules out fracture, can be strain of the tiny muscles between the ribs, the intercostal muscles. Some kind of trauma is usually involved, such as a fall, an auto accident, overexertion, etc.
These little muscles help with the movements of our chest when we breathe, so if they are injured, every breath can be painful.
Healing can be slow, but can be assisted by careful assessment of the rib cage for misalignments that put stress on the injured area.