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.
Continuity mistake: When Gomer drops the gun from the roof it lands on the ground directly in front of Andy and Barney, who are standing on the sidewalk beside the courthouse, but the area where they're standing looks different between the overhead shots and closeups, and their shadows change as well. Additionally, the gun lands with its handle to the left, but in the wideshot the gun is lying the opposite way, with its handle to the right.
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: Before Barney jumps out from behind the crate to surprise the two cops in the alley you can hear the director yell out "Don".
Visible crew/equipment: When Opie's up on the ladder feeding the birds, in the overhead shot of Andy and Barney the mic's cable running up under Barney's pant leg is visible.
Continuity mistake: When Otis makes up a poem about jumping rope, Barney lets go of the rope and Andy is left holding the other end while the rope is left stretched across the floor. One second later when Opie and his friend come running in, Andy is now holding the rope in his other hand all rolled up.
Ernest T. Bass Joins the Army - S4-E3
Visible crew/equipment: When the two men are carrying the window pane past Barney and Andy, we can see the reflection of a few crew members' legs.
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).
Visible crew/equipment: While Aunt Bee, Andy, Barney, and Gomer are on the porch talking about the band, just as Andy says, "We're talking about gettin' up a band concert by tonight," the chalk T-mark is visible on the floor by Andy's feet. (00:12:25)
Visible crew/equipment: Barney shows up with the concert sign while Andy's sawing wood, and when he and Andy talk about it, the shadow of the boom pole/mic is moving around on the ground, at the lower left corner of the screen. More of the boom's shadow is visible again, when Gomer shows up and they talk about Goober. (00:14:30)
Visible crew/equipment: At the end, when Dr. Breen tells the very tired group, "You all look as serene and relaxed as if you'd just finished listening to a pleasant band concert," in the next shot facing the porch, the shadow of the boom mic is moving around between the two windows above Clara and Andy, at the top of the screen.
Continuity mistake: When Opie shows Andy his arm with the chore list written on it, Andy uses his right hand to hold up Opie's arm, but in the closeup it's Andy's left hand holding up Opie's arm. (00:03:50)
Continuity mistake: At the start, when Andy stops at Wally's station Gomer is telling Jason about his Trout bait, then someone pulls up behind Jason's car and they're bumper to bumper in the wideshot, but in the exterior shot facing Jason there's no car behind him, but that car is back again before he pulls away.
Continuity mistake: In earlier episodes, we are told that Gomer can't do mechanic work. He just pumps gas and air: "There's no charge for air; air is free; we don't make no charge for air." But in this episode, Gomer has become an ace master mechanic.
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.