Can you sue someone if theygive you a contagious disease?

In the US, anyone can sue anyone for anything. The question is not whether you can sue, but whether you can win.

But to answer your question, maybe. This is a relatively new concept in the law, but there have been cases where people have won awards against someone who gave them a terminal disease. The most common example is a person who knows they are HIV positive, neglects to tell their partner, and the partner contracts HIV. The question of whether you can win depends on 1) the type of disease, 2) how it is spread, 3) whether the defendant knew he/she had the disease and was contagious, 4) the degree with which your life will be changed by the disease, and 5) your states' laws.

On the other hand, if you simply caught a cold from a co-worker, you are not going to be able to recover.