When Tony is riding the bronco, he is wearing a straw hat, but the real bronco buster in the stock footage is wearing a black hat.
When Jeannie is being introduced to Mr. Akins, the wide shot shows Akins holding his hat. In the cut to Akins's closeup, he is wearing his hat. Returning to the wide show, he is again holding his hat.
When Roger and Tony were signing up as cowboys, Roger asks "why don't we just sit next to her [Jeannie] in our uniforms?" Tony should have said "because Dr. Bellows would connect her with me." Instead Larry Hagman messed up the line and said "because she would connect me with Dr. Bellows"
Even though it was established at the beginning of the series that Jeannie was stuck in her bottle for 2,000 years before Tony released her, she constantly claims to have personally known historical figures from history: in this case, Genghis Khan, who lived in the 12th-13th centuries, over 1,100 years after she had been trapped in her bottle and more than 750 years before Tony rescued her.