To a certain degree. For most of the war, submarines spent the majority of their time on the surface (the invention of the snorkel, which allowed German U-boats to run their diesel engines while underwater changed this somewhat). Every sea-going vessel has small leaks, so submarines had bilge pumps to keep up with the slow influx of water.

When submarines dived, the pressure on the hull increased and the leaks would get worse. Not catastrophic leaks, but enough to make sure the interior of the submarine became humid and damp. Accounts of submarine life from every side constantly mention humidity, damp, mold, mildew, and terrible smells. Das Boot nails this one pretty much on the head.

If a submarine took damage (ex. the depth-charging shown in the film), leaks could get extremely bad. Riveted joints and/or welded seams in the pressure hull could fail, causing serious and potentially-fatal leaks. In other cases, the leaks might be smaller. Still, even a small leak under pressure could shoot a jet of water into a submarine. Here's a video of US Navy submarine damage control training https://www.youtube.com/watch?v=Rovel9AJXfA)