Continuity mistake: While Bud is napping/fishing in the skiff, Flipper grabs hold of the anchor line and tows the skiff. However, in Flipper's closeup, the anchor is missing from the line. Then it reappears when Bud pulls up the line and sets the anchor down in 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.