Generally speaking, the answer to your question is yes you could have an underlying disease and still have normal pulmonary function testing. I recommend that you see a pulmonologist
for this reason. The reason for this is that sometimes lung disease is at a very early stage and has no impact yet on your lung function. For that reason, the pulmonary function testing will return to normal even if an underlying lung disease at an early stage is present. The obvious next step for you is to get the CT scan that you already got. This will tell your physicians more than the x-ray about your potential lung disease. It is possible that you have no condition at all. There are many reasons to have a chronic cough and most of them have nothing to do with the lungs at all. You could have acid reflux or post nasal drip which are two of the most common causes of a chronic cough. If you're a smoker or you have asthma
those are two other chronic causes of cough. If the CT scan shows some abnormalities, then your doctors
will have to make a choice as to whether or not they want to observe your problem with further testing in the future or get a biopsy of the lung. This decision is not straightforward, and thus you should make sure that you get a second opinion if you are recommended to have a biopsy. You should see a pulmonologist for this issue.