My six year old son had a bad persistent cough all throughout the fall, from October through December. I took him to the doctor 3x in October/November and they kept telling me it was not in his lungs and he just had a cold. He missed a week of first grade b/c I didn't want him spreading a virus around. The cough would NOT go away. Then on xmas day, he threw up, kept coughing and then really started to wheeze. I got an 8pm appt at the pediatrician and by the time we got there, his pulse oxygen saturation was 90% which is apparently really really bad. They put him on a nebulizer right there, did 2 treatments in the office and were able to get it up to 95%. The doctor said if it had stayed at 94% or lower, he would have sent us to the hospital.
They put him on maintenance steroids for his lungs to reduce inflammation which was apparently there and had just been flying under their radar. It was very scary for me that night at the pediatrician's.
SO my point is, keep watching him and watching him. Keep taking him to the doctor so they can check his lungs and his pulse oxygen saturation, too, if you can convince them to do it (they check this with a little monitor to his index finger, its not invasive or scary at all). Just stay on top of it, so you'll notice any changes right away and can act appropriately.
You know your son better than anyone, and you know what is normal for him and what seems "not right." Don't be afraid to take him to the doctor even if they're telling you that its "just a cough" and will go away on its own. It's not normal for kids to have a persistent cough and it is a huge symptom of asthma and breathing trouble.
D.