Cats, like a lot of pets, cough for a variety of reasons, some of them completely benign. For instance, cats cough to expel hairballs; this is a natural process for them.
There is a less common version of kennel cough that exists for felines as well, called Feline Bordetella (also known as Bordetellosis or Bordetella bronchiseptica) and it can present a number of ways. The signs are very similar to feline herpes and feline calcivirus; that is, a runny nose, runny eyes, sneezing, coughing, and otherwise difficulty breathing, lack of energy, loss of appetite, and occasional fever.
The signs include: Sneezing, congestion, runny nose, cough, clear-to-colored nasal discharge, gagging , drooling, fever, loss of or decreased appetite, rapid breathing, nasal and oral ulcers, squinting or rubbing eyes, open-mouthed breathing, depression.
Viruses are, by far, the most common cause of upper respiratory problems in cats, accounting for 80%-90% of cases. Chlamydia and Bordetella are commonly found in animal shelter and catteries.