I heard Anna Maria Tremonti hosting The Current on the topic of boredom. The part that I found particularly interesting and relevant to mental health had to do with the state of physiological arousal that people tend to feel while claiming to be "bored". Rather then relaxed, bored people have a faster heartbeat and are in a state of agitation.
Below is a video of a trailer for a documentary made about boredom. It looks quite interesting. I'll be curious to watch the movie and see what kinds of conclusions are drawn about the connections between boredom and symptoms like depression and anxiety.