Could someone recover from a collapsed lung?

Almost certainly: If the collapse is a pneumothorax there is very high return to normal. If the individual has emphysema before the collapse, recovery is lessened.

At least a month: If you had a spontaneous pneumothorax and required chest tube drainage or a vats blebectomy operation you can probably resume near normal physical activity within 4-6 weeks afterwards on average.

Absolutely: The hole in the lung from trauma or spontaneous collapse can potentially seal in just a few days (like a scab). The lung tissue takes longer to fully heal, depending on type and extent of initial injury. If from trauma, associated chest wall injury (rib fractures, etc...) as well as chest tube site may take weeks to months to heal.