Answer:
No. For example, people in Alaska don't get more colds than people anywhere else. We do have more colds in winter than in summer, but not because of the cold (see more on why below). Cold weather conditions play no role except as mentioned below about absolute humidity levels in the winter. One of the expert scientists (Bill Nye) said that you can not catch a cold from being cold; you catch a cold from germs and being cold has nothing to do with it. Plenty of tests have been conducted proving this.
The old belief that freezing temperatures cause illness started before people knew about germs; however, it continues to be passed along to others as a legend today and is not an evidence based finding from studies - just based on anecdotes and incorrect associations.
It has been scientifically studied with double blind test groups and there was no difference found in the rate of infection with common cold viruses when the study groups were exposed to cold temperatures or heat via different methods. The results of those studies were peer reviewed. "No" has become the current most accepted answer to the question by scientists and medical professionals.
Then why do we get more colds and flu in winter and cold weather?
It had been long held that this was most probably due to school children returning to schools and people being in closer proximity indoors in winter where they could pass all their germs around more easily. One of the most commonly cited studies used as a basis for this hypothesis was the "Seattle Virus Watch", done by John Fox, Carrie Hall, and friends.
Another hypothesized explanation had been that our Vitamin D production is lower in winter due to less exposure of our skin to sunlight, and since Vitamin D improves the immune system's ability to fight off infections, our defenses are made weaker in winter with Vitamin D deficiency. Another commonly held belief was that in drier air our mucous tissues dry out and can crack, making the viruses more easily introduced to the body. Some combination of all these factors may be at play.
However, the most recent studies have all seemed to point more to the different absolute humidity levels in winter compared to those in summer. Cold and flu viruses like it dry. See the related question below, "Why does the flu have a season?" for more details about these recent findings.
Check out the discussion section for comments, anecdotes, and discussion.