Is it normal for a wet cough to linger after cold is gone? Plz Answer!?

Okay, so far no ones answered. I started getting a cough about a month ago. i had a minor fever and i was sick for a week then i got a fever of 102 and went to the doctor, the doctor said that as my first cold was going away my second cold came on so it was 2 separate colds. He told me to come in within a week if it didn't go away. I did and he suspected a sinus infection had spread to my lungs and caused an upper respiratory infection (i also had a fever of 103, and i thought i had pneumonia because i had it before) but i took antibiotics for a week and my cold symptoms have seemed to have gone away. But the cough seems to be staying with me. its been a week since the antibiotics stopped their effect. I am not coughing up phlegm anymore but i still can feel like i have it and it is a wet cough. Is it normal for it to last this long? If not would it probably be my Allergy induced asthma