Does sleeping fewer hours than needed cause common cold?
There have been some studies directly linking sleep deprivation to increased risk of catching a cold  ("Behaviorally Assessed Sleep and Susceptibility to the Common Cold Sleep". 2015;38:1353–9.).

Colds are caused by a family of viruses. There is pretty solid evidence that sleep deprivation has a significant weakening effect on your immune system. Given a weakened immune system, it would be more likely that you would catch a cold if you are exposed to one of the cold-causing viruses.