Will a partially collapsed lung re inflate itself over time?

Maybe it will: A partially collapsed lung is due to an air leak from the inside of the lung through the covering of the lung, out into the space between the lung and the ribs. The site of the leak has some damage, which will heal itself later. If only a very small amount of air leaked into the chest cavity, that air might go away (reabsorbed by the body) without treatment, and the lung will re-inflate.

Yes: If the cause for the collapse is evaluated, found and treated.