What happens if you try to fly with a partially collapsed lung?

SOB. Shortness of breath can occur if you do not have proper lung capacity and you fly at high altitudes without proper pressurization.
Depends on pressure. When flying with a partly collapsed lung, the leaked air in the chest (between the lung and rib cage) expands as the outside air pressure drops as the plane goes up. The expanding trapped air compresses the lung & heart, leading to shortness of breath, inadequate oxygen intake, poor circulation and death. Flying at low altitudes won't expand the trapped air much, but the plane might hit something.