Barney and the Governor - S3-E15
Deliberate mistake: In the closeup of the license plate on the governor's car, the words over and under the plate number are deliberately taped to be unrecognizable. The original words are "drive safely" and "North Carolina."
High Noon in Mayberry - S3-E17
Deliberate mistake: While Luke and Andy are in the house, Barney, Gomer, and Otis are hiding on the porch, and Barney has the idea to turn off the lights in the house, so he goes to the circuit box located beside the windows on the front porch and switches the lights off. The problem is that this circuit box is located on the porch only for this episode as a plot point.
Visible crew/equipment: After Andy and Barney have asked Otis if he's seen the billy goat, when it cuts to Opie on his bike we can see the two white tape marks on the ground where Andy and Barney stand.
Factual error: We know from 2 episodes prior (Man in a Hurry) and generally mentioned throughout the show, the only filling station in Mayberry is Wally's Filling Station. In this episode Mayor Stoner is accused of only putting the new underpass in Mayberry because it will be next to Mayor Stoner's brother's filling station.
Continuity mistake: In the closeup of Barney's yearbook picture his name is listed as "Bernard Milton Fife," and Andy even reads it aloud, but in the previous episode from season 1, "A Plaque for Mayberry" Andy says that Barney's middle name is Oliver.
Character mistake: When Andy and Barney are looking through the yearbook, Barny's middle name is Milton. But Barney says many times throughout the series that his name is Bernard P. Fife.
The Great Filling Station Robbery - S3-E22
Visible crew/equipment: When Gomer walks into the courthouse with the envelope and Barney takes the photo out, the shadow of the boom pole is moving around on the door and wall to the right.
Aunt Bee's Medicine Man - S3-E24
Continuity mistake: The pillow on the right side of the couch is there while Opie is talking to the Colonel. When Andy walks around after Opie leaves, it has suddenly disappeared. (00:16:00 - 00:18:00)
Aunt Bee's Medicine Man - S3-E24
Continuity mistake: Andy and Barney are playing checkers near the end of the episode. Andy can be seen making 2 moves in a row, skipping Barney's move.
The Darlings Are Coming - S3-E25
Visible crew/equipment: During the scene where the Darling's truck has overheated, Andy walks up from the back of the truck. As he comes into view you can see what appears to be a microphone cord hanging out from under his pants leg.
The Darlings Are Coming - S3-E25
Visible crew/equipment: After Charlene tells her father that she just wants to look at the pretty man (Andy), when she gets back in the pickup truck we can see the reflection of the moving boom on the windshield.
The Darlings Are Coming - S3-E25
Continuity mistake: When Briscoe gets into his truck, his hat is wet from dipping it (5x) in a horse trough. The camera cuts away to Andy and immediately back to Briscoe, whose hat is now dry. Also, "she's always hood for 11 hat fulls," yet Briscoe only puts 5 hats of water in his engine.
Deliberate mistake: After Barney buys the car from Mrs. Lesh, he and Andy are standing at the steps as he tells her not to look back at the car, and the next shot of Andy and Barney at the front door is the same shot from a few moments prior, but it's been flipped so they would be looking in the opposite direction. (00:09:30)
Visible crew/equipment: While Andy and Barney are hiding behind the car, when Mrs. Lesh hits Jake in the stomach with her umbrella, we can see two curved chalk actor's marks and two chalk T-marks on the floor. (00:22:10)
Revealing mistake: In the scene where Andy starts reading Opie a story, watch when the first woman rings the doorbell. As soon as Andy opens the door you can see that the set all of a sudden gets a lot brighter, as if they turned more lights on.
Character mistake: Mr Darling says there are only 5 strings on the guitar he hands to Andy. You can see all 6 strings when the camera shows Andy playing Dooley.
Continuity mistake: In this episode, Ernest T writes Charlene a love letter. In a later episode when Ernest T goes back to school, it is revealed that he can neither read nor write.
Plot hole: When Barney and Andy are on the street watching the window painter, Andy looks over and says "I believe that's Briscoe Darlin's truck!" - but when they were driving up to his place the next morning, they both got out of the car and were wondering where their house was they heard music playing. The next scene shows them pulling in - seems that if he was able to recognize the truck in town, he would have been able to recognize the truck and they could have drove around looking until they found it?
Continuity mistake: At the end, when Andy walks in on Barney showing Gomer how to draw his gun, Gomer draws his gun out of the holster which sends it flying behind him, and in the closeup the bookcase glass shatters, but in the wideshot when they all turn to look the glass is still perfectly intact.
Visible crew/equipment: When the partners of the two men locked up in the jail drive up to the courthouse and park, the reflections of two crew members are visible on the car's vent window.
Answer: Those "boxes" are valences that used to be very common, before air conditioning. They allow for windows to be open during rain storms. They permit air circulation, without letting the rain in.