Could being cold lower your immune system or something, making people more likely to catch a cold?

Not at all: You always heard your mom say "you're going to get sick if you don't bundle up" has been shown to not be accurate. Exposure to cold temperature in typical urban or suburban settings would not down regulate immune responses. However, prolonged exposure to extreme cold temperature even for short periods of time can cause a host of circulation issues affecting all organs and in turn can cause ills.