The common cold does often occur with a cough. If you are currently experiencing cold symptoms or are having recurrent cold symptoms--I would suggest you see a primary care doctor
who could help answer your question.
The common cold refers, in general, to a viral infection that causes inflammation of the sinuses, nose and lungs. This inflammation often gives the symptoms of a runny nose, stuffed up sinuses and cough. The reason we cough is because the inflammation in the airways causes irritation in the airways. Our body has irritation receptors in our airways, that if they sense irritation, they trigger a cough. The cough is a defense mechanism that we have developed to clear the airways so that adequate air flow can occur. That is to say, that the cough is a symptom and not the disease itself. In fact, the cough is a defense mechanism. Therefore to cure the common cold we cannot simply just stop the cough...we need to figure out how to treat the cause--which is the virus. Unfortunately, we are not very good at identifying viruses, let alone treating them. We in fact can, with some success, suppress a cough fairly well with various medications such as tessalon pearls or other suppressants.
The common cold is caused by a virus which often results in a cough. Talk to your primary doctor if you have further questions.