While the cold doesn't cause illness, it seems to have the potential to reduce immune response which can then result in more illness than if one hadn't gotten cold. From Wikipedia's page on the common cold:
While colds are caused by viruses and not cold temperatures, there is some controversy over the role of body cooling as a risk factor for the common cold; the majority of the evidence suggests that it may result in greater susceptibility to infection.It may have been coincidental but I fell ill with colds a day or two after both times I got caught out in the cold with inadequate clothing this winter so I had to look this up.