Would coughing too hard strain a muscle in the chest?

Yes: Coughing can be associated with muscle strain, rupture and even broken ribs. Most of the time cough is a good mechanism to keep the lungs clear of secretions; if your cough is too forceful you can sustain muscle injury. Sometimes health care providers will give you medications to decrease the intensity and severity of cough for this reason.

Possible...: Once a cough lasts for a few days or weeks, it is possible to have musculoskeletal problems like pulled muscle, inflamed cartilage and even rib fractures. There is no specific therapy for these except symptom control and measures to stop the cough.