The particular case that Hygiene99 mentions (about turning back because the family had a small child on the ship) is a rare exception.
Normally, cruise ships sail when scheduled, and they keep going. It's the passenger's responsibility to be at the port on time. And it's the passenger's responsibility to travel to the next port to join or rejoin the cruise if they get to the ship too late. (If the cruise line arranged the flight and that flight fails to get the passenger to to port in time for boarding before the ship sails, then the cruise line pays for the tavel to the next port.)
But consider this... If you miss the sailing of the Eastern itinerary Disney Magic on Saturday, your first chance to join the cruise is on Tuesday in St. Maarten. You've just missed half the cruise.
The captain should delay sailing if a Disney bus is late arriving a Port Canaveral or a ship-sponsored shore excursion is late returning (both highly unlikely). The captain may choose to delay sailing if a substantial number of passengers are not on the ship at the time scheduled sailing time -- for example, if airports in the Midwest or East Coast are closed in the morning, and many flghts are delayed until the afternoon.
Bottom line: Allow plenty of time to make it to the ship. Fly to Florida on the day before sailing. At ports, be back on the ship at least an hour before the scheduled departure. Then you won't have to worry about what your options are.