A runny nose on your cat can mean any number of medical problems. Please do not give your cat any human medicine, such as cough or cold medicines, as these can poison your cat. Observe your cat for other symptoms of distress or unhealthiness, such as coughing, lack of appetite, inability to use the litterbox, lethargy and crying. Most runny noses in cats will go away on their own in a few days. If you are concerned about your pet's health, please contact your veterinarian.