I've googled the condition many times, looking for answers. I found one article by a veterinary that pretty much covers all bases: Click HERE to read it in its entirety. It explains why the shelter is not at fault. Here are a few excerpts:
(California) State law requires dogs usually be held in a shelter six days before they can be adopted. During this time, the stress of shelter life and exposure to airborne bacteria make shelter dogs prime candidates for kennel cough, even if vaccinated. Although the symptoms can occur in three to five days from the time of infection, some veterinarians report the actual coughing does not begin for seven to 10 days. Because even the fastest acting vaccines don’t provide protection for at least four days, shelter dogs may often look perfectly healthy at the shelter and start coughing almost as soon as they get home.
Kennel cough is usually what veterinarians call “self-limiting.” That’s a fancy way of saying you probably don’t need to take your kennel cough dog to the vet. The problem will resolve all by itself in 10 days to three weeks. Although it is caused by a bacterium, antibiotics are generally not needed.
If you see a kennel dog you want to adopt, don’t be discouraged even if they are coughing. Kennel cough is easily overcome. Once a dog is taken from the stress of a shelter and given a loving home and some time to heal, you’ll soon forget about the cough. But the dog will always remember that you gave him a home when others wouldn’t.