Yes. They can catch a cold and I'm talking besides Kennel Cough and all the other little nasties. My first dog would get one every couple of years. He looked just like a human with a cold; snotty nose, runny eyes, slightly off his feed, a bit grumpy, cough, and sneezing. Every time, we'd rush him off to the vet and every time the vet would give him a once over and give us a prescription to give him lots of love and a bowl of homemade chicken soup or light broth. We were told every time that it was just a cold. Within 3-5 days, he'd perk up and be his normal self. Finn got one right after we moved to MD, but he had a nasty case of Gastroenteritis that lasted for what seemed like forever, but was barely a week before he was completely back to normal and the cold hit not even two weeks after that. The vet told us it was just a cold and his momentarily weakened immune system couldn't fight it. He was fine in a week or so.
I would take the pup to the vet to make sure it isn't Pneumonia, KC, or Canine Influenza, but yes, they can get a "cold". I was always told lots of rest, quiet, and low light. (first dog's vet swore dogs could get headaches and become slightly light sensitive. Not sure if it's true, but Sebastian did seem more comfortable when he was in a low light room and the same went for Finn.) Also light, system easy meals like broth with chicken, boiled ground beef in rice, sweet potatoes to help with troubled bowels, and stuff like that should help your pup to come around after the all clear from a vet.