Wet Cough that is lasting a while!?

Okay i recently had a sinus infection, that the doctor said might have gotten some stuff into my lungs that irritated them (i have asthma) so i had a wet cough, so i took antibios and its been about a week and a half and i still have that same wet cough. I do get allergies at random years. But i usually get prednisone and the doctor would not give me that this time. Is it normal for a cough to last this long, if not do you think its my allergy induced asthma