The Taylors in Hollywood - S6-E8
Revealing mistake: While Andy, Aunt Bee, and Opie are at the Hollywood studio watching a scene being filmed, when Andy mistakenly stands up and interrupts the filming we can see the wall and shelves behind the director, and it should look familiar because it's the wall and shelves from Floyd's shop in Mayberry.
Andy and Opie - Bachelors - S3-E4
Revealing mistake: After Andy and Opie choose between chicken and pot roast using a nickle, when Opie gives the nickle back to Andy the window over the sink is missing - the fake window wasn't put up on the set wall for this shot, but when the food burns the window and curtains are up there as usual.
My Fair Ernest T. Bass - S4-E17
Revealing mistake: At the start, when Ernest T. throws the rock at Mrs. Wiley's window and it breaks, the area around the hole flutters in the breeze - it's obviously not a glass window used for the stunt.
Aunt Bee's Invisible Beau - S5-E27
Revealing mistake: When Clara and Aunt Bee are talking in the kitchen, in the closeups facing Aunt Bee we see the counter, the cabinet, and the wall, but what we don't see are the curtains, the window, or the windowsill, which all should be on that wall. The fake window wasn't placed on the set wall for Bee's closeup.
Revealing mistake: At the end, in the shots of the kids dancing in class, the width of the wall between the shelf/coat hooks and the perpendicular wall with the door has bizarrely increased by 1-2 feet, but that extra width is gone in previous and following shots. Presumably they needed extra space for all the kids to dance so they moved that left wall.
Revealing mistake: When Andy tells Opie not to drop by the courthouse anymore after school, in Andy's closeups the shots are flipped - note the part in Andy's hair, the top of his badge, and his backwards buttoned shirt.
Revealing mistake: While Goober is up in the tree with the fake owl, when Andy and the Darlings leave the house they are standing on the porch, and after seeing the 'second omen' the Darlings leave the porch and head to their pickup truck, but what's funny is that we can see their reflections in the window behind Andy as they all walk off camera, turn around and just stand there, and we can still see those actors' reflections in the shots facing Andy and Aunt Bee even as the Darlings supposedly drive off.
Revealing mistake: When Barney pulls the jail cell window out of the exterior brick wall, as it's being yanked out we can see that the brick wall is only thin drywall during the stunt, as well as the fact that the wall around the breakage differs significantly between the closeup and wide shots.
Aunt Bee the Crusader - S4-E15
Revealing mistake: Toward the end of the show the rooster, Bo, is drunk. If you look closely you can see the two strings pulling the rooster's neck sideways.
Revealing mistake: During the holdup at the bank, while the bank robber's holding Asa's broken gun, when Andy motions to Asa to duck out of the way in the closeup we can see the two thin wires to the left and right pulling the gun apart.
Revealing mistake: During the scenes with Opie and Dolly in Walt's barn, the horse is wearing a halter with a ring under the adjustable chinstrap, and in every closeup a thin wire (like picture wire) can be seen attached to the ring as it's being pulled from offscreen to have Dolly turn her head away from Opie. Then later when Walt returns and tries to feed Dolly, once again a thin wire attached to Dolly's halter is pulled from offscreen, so she will turn her head away from Walt. (00:05:55 - 00:18:20)
Revealing mistake: When Andy arrives at the studio for the publicity shots, while he's speaking to Mr. Michaels note that behind them we can see the Taylors' Mayberry kitchen cabinet/counter and a wall with their distinctive gold/silver design wallpaper. (00:03:50)
The Taylors in Hollywood - S6-E8
Revealing mistake: When Andy, Aunt Bee, and Opie are visiting the studio set while the scene between fake Aunt Bee and fake Andy is being filmed in a fake Taylor living room, note that behind the real Taylors we can see bits of the interior of their Mayberry house scattered in the background offstage. Some things to look for include their living room's stone wall with the mantle, their two reddish chairs (one with the fringe at the bottom), and a wall with their distinctive gold/silver design wallpaper. It's particularly funny when Opie looks at the "fake" living room set and comments, "Is that our house? It sure looks different!" while we the viewers can see the "real" Taylor set behind him. (00:20:35)
Revealing mistake: While Barney's explaining to Andy his new interest in realty they both walk to the back room, and just as they pass through the doorway the camera tracks too far to the left, so we can see the edge of the set wall and the rest of the back wall is missing. (The following shot is from the direction of that missing wall).
Revealing mistake: When Andy, Barney, Gomer, and the kids are at the campsite in the woods, during the daytime scenes the only shadows that should be seen are the ones being cast by the sun, but there are shadows being cast in different directions due to the set lighting.
Revealing mistake: During The Manhunt in the woods, Andy, Barney, the trees, etc., cast long shadows due to the overhead sun, but the problem is that we also see Andy's and Barney's shadows being cast in another direction as well, due to the set lighting on location.
Revealing mistake: When the bus pulls up for the Taylor family to leave for Hollywood you can see behind the bus a California freeway with cars moving in both directions. I don't think that Mayberry had a freeway right outside town. The scene is fixed when they return in "Aunt Bee on TV." There has been a tree put there so the road wouldn't show.
Ernest T. Bass Joins the Army - S4-E3
Revealing mistake: When Barney picks Andy up and they get back to the courthouse, they find Ernest T out of the jail cell again and breaking more windows, and in the shot of Ernest T leaping in the air we see it's actually a flipped shot - note the backwards "snack bar" in the window (that store is later when Ernest T tries to break the new glass for the courthouse).
Revealing mistake: The jailed criminals take the key and open the jail cell door. The problem is the dead bolt never moves with the key, showing the door was never locked to begin with.
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.
Answer: It was a joke. They were supposed to be dumb hillbillies.
Both. According to later interviews with both Andy and the Dillards band (the real name of the band that played the parts of the Darling children, of which Charlene (Maggie Peterson - Mancuso was also a member) the reason the Darling boys never had any lines was because they would've had to have been paid if they spoke. In return for appearing without lines or pay, Andy promised to get as much of their music onto the show as possible which, he did. This obviously gave them a lot of exposure.