According to information from the Mayo Clinic, exercising if your congestion symptoms take place "below the neck" is not recommended, as physical activity can potentially exacerbate the problem. However, if you are training for a particular event or you are just the kind of dedicated individual that never misses a workout, there are some steps you can take to minimize the affect of your chest congestion, allowing you to still train at a reasonable level of intensity even while ill.
Exercising with Chest Congestion
Assess yourself realistically prior to engaging in exercise while suffering from chest congestion. If you are affected with a severe, deep-lung cough or are having difficulty making it to the restroom, let alone the gym, you are better off skipping the workout entirely. On the other hand, if you have transient or mild symptoms that vanish with nothing more than a dose of cold medicine, you can probably still train safely.
Choose low to moderate intensity exercises that will not elevate your heart rate or breathing to unreasonable levels. If you are already suffering from chest congestion, the last thing you want to do is trigger a coughing fit by engaging in fast running or heavy lifting--the kind of exercises that will leave you gasping for breath afterwards. Choose mild activities like brisk walking or training of smaller muscle groups like the arms, shoulders, and calves. While you might be winded after a particularly heavy set of squats or deadlifts, the same is not necessarily true of calf raises, so stick to exercises that you know you can perform without undue strain on your body.
Call it a day if your symptoms escalate while training. The goal of training is to improve your health, not impair it. If you feel that the training session is making things worse you are better off quitting on the spot than forcing yourself to complete it just to regret it tomorrow when your symptoms are even worse. Maintain constant awareness of your physical state to avoid pushing it too far beyond the limit while exercising with chest congestion.