This is a fairly common problem for children at this age. There are several possible diagnoses that can cause the repeated episodes of coughing. Here are some and ways to evaluate them.
· The most likely cause is a sinus infection. This causes coughing that goes away with antibiotics. A CT of the sinuses is the only way to make the diagnosis. While this is a very safe test, it is not so easy to do on a 2½ year old.
· Bronchiolitis is NOT the cause. It is not repetitive in the way you have described. Usually bronchiolitis causes wheezing with the coughing and really does not respond to antibiotics, as it is a viral infection.
· Asthma is a possible cause. It is usually not an allergic problem in children this age. It is more likely due to viral infections. Therefore it is fairly common in the winter. It also is made worse by sinus infections. Most of the time, but not always, the child wheezes as well as coughs. If the cough is due to asthma, it should respond very well to a short course of oral steroids.
· General allergy does not cause this type of problem in a 2½ year old.
It sounds like your pediatrician is frustrated and seeking help. A good pediatric allergist or pulmonologist should be able to sort this out for you easily.