This table shows seven types of feline coughing with suggested causes. Remember a cough is a symptom of an illness. The information comes from four American veterinarians: Debra Ekdredge DVM, Delbert Carlson DVm, Liisa Carlson DVM and James Giffin DVM. Coughing is a reflex action instigated by an irritant in the bronchial tubes or perhaps the throat. Coughing creates more coughing. The cycle has to be stopped.
A specific diagnosis can be obtained by examining cells recovered from the trachea. This is carried out with the cat under light anesthesia by a vet. This is called a transtracheal washing. X-rays are also taken.
Below are 2 more pages on the cat cough that might be useful.
An asthmatic reaction is the most likely cause of feline coughing