A team from Carnegie Mellon University found the risk was trebled compared with those who slept for eight hours or more a night.
It is thought that a lack of sleep impairs the immune system and the body's ability to fight off the viruses that cause colds and flu.
The study appears in the journal Archives of Internal Medicine.
Previous research has suggested that people who sleep seven to eight hours a night have the lowest rates of heart disease.
However, there has been little direct evidence that getting a good night's sleep can help ward off a cold.
The researchers studied 153 healthy men and women with an average age of 37 between 2000 and 2004.
Each was interviewed about their sleeping habits over a two-week period.
They were then quarantined and given nasal drops containing rhinovirus, which causes the common cold.
For the following five days the volunteers reported any signs and symptoms of illness, and had mucus samples collected from their nasal passages.
And 28 days after exposure to the virus, blood samples were taken from each volunteer so tests could be carried out to see if they had developed antibodies to fight infection.
The less an individual slept, the more likely they were to develop a cold.
The quality of sleep also appeared to be important. Volunteers who spent less than 92% of their time in bed asleep were five-and-a-half times more likely to become ill than those who were asleep for at least 98% of their time in bed.
The researchers believe that lack of good quality sleep disturbs regulation of key chemicals produced by the immune system to fight infection.
Professor Ron Eccles, director of the Common Cold Centre at the University of Cardiff, said sleep and the immune system were closely linked.
He said: "The immune system may control the sleep-wake pattern and lack of sleep or sleep disturbance may depress the immune response to infection.
"I do believe there is enough information on this to indicate that lack of sleep or sleep disturbance will reduce our resistance to infections such as colds and flu."
Dr Adrian Williams, director of the Sleep Disorders Centre, at St Thomas' Hospital, London, said the study echoed previous work in animals suggesting sleep had an effect on immunity.
Dr Neil Stanley, a sleep expert at Norfolk and Norwich University Hospital, agreed that previous research had shown that poor sleep impacts on immunity, but he said there was little data on its effect on specific infections, such as colds and flu.