Somewhat true. Being cold has no direct impact on getting a cold. A cold is caused by by a rhinovirus. Being cold has no impact on your immune system or the virus. The reason more people get colds in the winter IS because our houses become cultures for the virus. We're inside a lot with the windows closed which allows the virus to be spread more easily. But this is because we're in close quarters, not because it's cold outside. In fact, you'd be less likely to catch a cold if you stayed outside in the winter.
Pneumonia is an infection of the lungs. It is only associated with a cold because sympotms can be similar in the initial stages. It can be caused by bacterial, viral, or fungal infections, but not from being cold. It can also becaused by inhaling certain chemicals. Taking a bath will not give you pneumonia, unless the liquid is a mixture of ammonia and bleach, or certain other toxic mixtures. The flu is also caused by a virus.
In fact, the only thing that I can think would happen if you get too cold are things like frostbite, shock, death, etc. But you can rest assured of two things: Your fan can't cool you off that much, and if any of those things happened, you'd know.
Disclaimer: I am not a medical professional of any sort. This information is from a variety of sources and is true to the best of my knowledge. I provide no guaruntee that it's accurate.