We had our Yorkie Tiger for about 3 weeks and he was diagnosed and treated for "kennel cough" by our veterinarian. (your puppy's cough sounds a lot like what Tiger's did which sounded like a hacky cough). After the initial course of antibiotics, the coughing slowly came back again. One week after Tiger finished his antibiotics he had an accident......he jumped out of my 9 year old's lap while she was holding him at the computer and he shattered his elbow. While prepping Tiger for surgery they noticed the coughing and did a chest X-ray which showed that he actually had bronchitis. Surgery to repair his shattered elbow was post-poned for 2 days to get him healthy as a different antibiotic was prescribed. If your puppy does have kennel cough and it doesn't go away - take him back and ask for another antibiotic :