The common rule of thumb is YES you can run if your symptoms are “above the neck” and NO you shouldn’t run if our symptoms are “below the neck.”
Above the neck symptoms include runny nose (who doesn’t have a runny nose all through the month of February?) sneezing, and sore throat. If the symptoms of your cold are below the neck, such as chest congestion, hacking cough, fever, muscle aches or vomiting, you will do more harm by exerting yourself than good. If your body is tired, it should rest.
But there is more to add to that simple rule. Researchers have noted that light to moderate activity may boost your immune system. But we do have to judge whether our workouts are really “light to moderate,” especially because researchers have also noted intense workouts can lower your immunity and make the illness take longer to resolve.
Coughing is not easily categorized by the above/below the neck rule. Some colds can leave a cough lingering for weeks and is just a residual effect of the cold. If your cough interferes with your breathing during your workout, then slow down. If your cough seems exacerbated by the exercise, you may have to put off resuming workouts until you are rid of that symptom. If your cough is worse outside, consider treadmill running for less irritation to your throat and chest.
Another exception to the above the neck rule is sinus infections. The pressure and infection is indeed above your neck, but running can exacerbate the infection and can lead to pneumonia.
Bottom line: Take a few days off when the most intense symptoms of the cold make you miserable. Your body needs to rest and heal. Then you can assess how this cold is developing. If, in a few days, you are left with just a runny nose, tickly cough or tiredness, a light to moderate run is okay and you can ease back into training.