Karla is right. Have you contacted the kennel and asked if other dogs ate affected?
We have a boarding kennel here that is extremely vigilant about sanitizing, making sure all dogs are vaccinated, etc. few weeks ago, they called and notified a neighbor that had reservations to board her dog that they had an outbreak of kennel cough. Facility was completely shut down and sanitized and reopened. I have to give them credit for notifying their clients!
If the cough continues, best to consult the Vet. Hopefully, it won't get too bad but has to run it's course. Please make sure you keep your dog home so as not to spread it.
http://www.dogs4dogs.com/blog/2012/03/21/bordetella-does-your-dog-really-need-the-kennel-cough-vaccine/
"Is it possible he could still get the disease?" YES......The Kennel Cough Vaccine covers about 2 strains of Kennel Cough...There are approximately 30 something different strains which make the shot a JOKE!!!! Karla