What exactly does it mean if you still have a bad, deep cough a week after a cold?

Cough after a cold: If a cough continues after a cold this could represent a sinus infection with cough from post nasal drip or less likely a pneumonia. If the "cold" was really pertussis (whooping cough), the cough may continue for a few months. With asthma, there is lingering cough after colds. This may/may not be accompanied by wheezing and shortness of breath. If cough persists over 4 weeks see a doctor.