When Tony comes home and walks in the door, he asks Jeannie to come out of her bottle. On the cut to the interior of the bottle, it is shaking and she looks upset, as if Tony is picking up the bottle, but the bottle is sitting still on the desk.
On the cut to the NASA interior on Tony's door, it reads, "MAJ. ANTHONY NELSON," but when Dr. Bellows enters the office , the sign on the door just says, "ANTHONY NELSON."
When Jeannie pops in to the audience at the talent show, she can't be invisible since the man next to her notices her when she hands him his hat. But, no one in the rows behind her notices her suddenly appearing.
When Tony gives back Roger his hat after cracking and mixing the egg, you can clearly see that inside of the hat is covered with a lining of clear plastic.
In order for this plot to work, Dr. Bellows needs to catch Tony in peculiar situations like floating in the air but he has to walk into the house when this happens. His house door is always unlocked and Dr. Bellows never rings the doorbell or knock on the door. The audience might wonder at some point that he would start locking his house door so that Dr. Bellows can't walk into it without ringing the doorbell.