Many studies say that negative emotions play an important part in emotional and mental well being, Depression helps to alter our consciousness and give us a different perspective on things that are troubling or concerning to us. Anger and other bad feelings draw attention to problems or important matters that need our attention.
Sometimes negative feelings result from physical pain, or cognitive dissonance, and other times they result from legitimate concerns.
For detailed information on negative feelings and well being, please refer to the following article in Scientific American.