The answer is sometimes, if you're damn lucky.
Last weekend a passenger fell from a Norwegian Cruise Line ship leaving New Orleans and survived; the crew saw him go over and immediately went after him. After being treated by NCL medical staff he disembarked for additional medical care.
In 1996, shortly after a Carnival ship sailed from San Juan, a drunken 23-year-old passenger fell off the ship. Part of a group celebrating Spring Break, the man decided to urinate off the side of the ship and climbed over a 4 1/2 foot high railing into a restricted area. He fell 76 feet into the 850-foot deep harbor.
Both the Coast Guard and Carnival staff immediately commenced a rescue operation with boats and helicopters after closing the harbor. Despite hours of searching they never found him. The passenger swam four miles back to San Juan and was found sitting on the beach, banged up a bit but sober. While he was in the water for 10 hours he was in relatively good condition.
The men who survived had two things in common: other passengers saw them go over and immediately notified the crew. And, they were very, very lucky. Other passengers have disappeared from cruise ships for unknown reason and never seen again.
Kitty Survives: My brother and sister in law were aboard a Windjammer cruise when the ship's cat fell overboard after being perched on a railing calmly watching passengers party. Crew members raced to a boat, rowed back to the spot he went overboard and pulled the terrified kitty from the water.
When they arrived back at the Windjammer boat, however, they were met by a Captain furious they had conducted a rescue operation without permission (he wouldn't have given it, not for a cat). And, he announced that the cat would be disembarking at the next port of call.