Deliberate mistake: After Bud overhears the plans for Flipper, he decides to hide his dolphin. When he boards the skiff, there are two flipped wide shots before they head out. Note the reversed dock, launch, and "Whaler" on the skiff.
Continuity mistake: When Flipper guides Rick to Bud's location, Rick carries Bud to the launch and hands him to Mr. Enfield, with Bud's head at Enfield's left arm. It cuts to the next shot, and now Bud is lying in Enfield's arms the opposite way, with his head at Enfield's right arm.