The last scene in the film shows Sandy riding Flipper in the water. When it cuts to underwater shots, Sandy is wearing blue flippers but the above water shots clearly show him barefoot.
When Sandy and Kim are feeding Flipper off the dock, Sandy is barefoot. Then Sandy climbs down the ladder onto the platform, and suddenly he's wearing a pair of black Chuck Taylor hi-tops, but is barefoot again when he goes back up on the dock.
When Sandy is looking for the injured Flipper, Sandy swims under water. He then gets back in the boat and his hair is dry.
When Sandy and Kim are snorkeling just before Flipper is injured, Kim is wearing striped shorts, except for a shot where she is wearing a dark swimsuit with no shorts.
Throughout the movie, but especially during the approaching hurricane scene, the stock footage conflicts with the scene footage. For example, Sandy and Porter are on the boat with flat seas and blue skies, but the stock footage shows dark gray, storm clouds, which would also yield higher waves on the ocean. When Porter returns from his trip to Ocala, the skies are blue and bright, but then there is a cut to stock footage of the beach at dusk.
The roman numeral copyright date at the end of the movie shows MCMXLIII which would be 1943. For 1963 it should be MCMLXIII.
A friend of the family stops by to say that Porter will be home "tonight." He then further clarifies it will be around midnight. However, when Porter comes home it is daytime.