That would be USS
Squalus. The crew didn't swim to the surface, but were rescued by a team led by legendary underwater expert Charles 'Swede' Momsen.
http://www.history.navy.mil/faqs/faq99-1.htm
And the water in this case was pretty deep - about 75 meters.
On the other hand, when USS
Tang accidentally torpedoed herself, some of the crew trapped below did manage to escape, using a breathing device also invented by Momsen. The depth that time was 180 feet - about 55 meters.