From my research it seems to be inflamed cartilage of the sternum, caused primarily by lifting heavy weight.

Basically I wake up in the morning, and sometimes throughout the day, I feel a pressure on my chest. When I twist my torso the pressure/pain increases.

It is alleviated when I lean back and stretch, and hear a satisfying pop in my chest. Very weird and kind of gross, and probably not good for me.

They call it "costochondritis", costo being the inflammation of rib and sternum fascia, caused by persistent tension of the upper thoracic muscles.

Most of the people I see on the internet with this problem have some sort of weight training routine they follow. It makes it hard for me to do dips.

Fish oil, daily stretches seem to make it feel better.