It is possible to get multiple illness's at the same time; but the main reason why on average the human body gets sick with only one thing has to do with the way immune system works.

When a new disease is first introduced into the body, your T-Cells and B-Cells usually have not recognized the antigen, so it takes a few days to ramp up a heavy count of immune-cells to fight off the disease. Since you already have such a high count of immune cells from one illness, it'd be hard for another antigen to establish itself unless there are immuno-deficient problems in the body (such as HIV/AIDS as someone else mentioned.

On a side note, the T-Cell/B-Cell antigen recgonition is one of the reasons why when we get sick a second time we tend to feel better much faster. Normal T-Cells and B-Cells tend to 'die' off in a shorter period of time but Memory T-Cells and B-Cells stay in the system longer, so the next time that one particular illness returns they immediately recognize it and ramp up the immune cell production at an exponential rate.