You can hear mothers everywhere calling their kids back inside from playing outdoors in the cold, shouting that they will catch their death. Being wet and cold can certainly make you feel lousy, but it doesn't make you sick. Even when scientists put cold viruses directly into people's noses, people who were chilled were no more likely to become ill than those who were warm and comfortable. We may get sicker during the winter months, but that's not because we are out in the cold.
Cold and flu viruses are transmitted from person to person through close contact and we all tend to congregate more indoors when it's cold or rainy outside. Close up the house and crank up the heat (or air-conditioning) and you dry out the natural protective mucous in the nasal passages making it more conducive for viruses to sneak into the bloodstream.
Actually being a little cold might provide some benefits. One study found that cooler temperatures actually stimulated parts of the immune system.