Other mistake: When Hank runs into the building to hide and lays down in the square of moonlight on the floor, he looks up at the window and a couple of clouds pass by. The clouds move too quickly, and when Hank is shown again, no shadows from the clouds move across him.
A Beer Can Named Desire - S4-E6
Continuity mistake: Toward the end of the episode Don Meredith takes Hank to the Superdome to try throwing the football into the Alamo beer can. After Hank makes the throw they walk to the football rack and Don picks up one of the footballs from the rack. We can see the top shelf has two footballs with the laces on the ball not visible. As they leave we can see on the same angle, there are three balls on the top shelf with the laces visible on all three balls.
Hillennium - S4-E10
Continuity mistake: In the episode we see the door to the boiler room (where Peggy's computer is) the door opens outwards into the hallway. Later in the episode she comes in to the room to check on some of her printings she was doing overnight. In this scene she opens it inwards. This wouldn't be possible as the door has a knob meaning it wouldn't be a two-way door.
De-Kahnstructing Henry - S3-E13
Continuity mistake: During the scene where Hank is talking to Kahn in the restaurant bathroom you can see a black rubbish bin below the hand dryer next to the entrance of the bathroom. About halfway into the scene Kahn starts looking into the mirror and the garbage bin has disappeared. Throughout the rest of the scene the bin never reappears. (00:18:20)
Continuity mistake: When Hank's trying to convince his dad to let the Vietnam vets join the VFW, the table behind him with the bowl of fruit and the vase of flowers on it disappears between shots. (00:12:45 - 00:13:15)
Continuity mistake: In the scene where the mechanic is checking Hank's car, there is a guy in the car who is sitting on the right side with his hands on the wheel. The problem is the steering wheel is on the left side of the car. In the next shot the steering wheel and the guy return back to the left side of the truck.
Continuity mistake: Whilst Hank is talking about the price of popcorn at the movies Bobby & Connie come up to Hank and there is a blue poster next to Bobby of a woman holding a gun. In the next shot you can see the blue poster is now next to Hank despite a red & orange poster next to Hank in the previous shot of him.
Continuity mistake: When Hank & Peggy are discussing what film to see, Connie & Bobby were to the right of Peggy who is right next to the truck. In the next shot Hank & Peggy are getting in the truck and Bobby and Connie are already there. Yet we would have had to have seen them go behind or in front of Peggy to get into the truck.
Continuity mistake: When Peggy is insulting Bill to try and get him to sell Metalife products again, he has a beer in his left hand. In the next shot when he tries to convince Peggy to give him another chance the beer has disappeared even though he never put the beer down. They were outside on the front lawn so there was not enough time for him to hide it or place it somewhere else on his front lawn.
Continuity mistake: When Bobby and Hank are in the pet store looking around, there are some red & white bags of dog food. Hank then asks the store clerk how much it costs to put a photo of Ladybird on a mug, and the color of the bags has changed. One of the bags is orange and white, whilst another is yellow and white.
Continuity mistake: When Dale is talking about getting the silver medal, you can see that on the wall, near the bed, there is a wind catcher. Once Nancy says "If you say so, I was pretty drunk", the wind catcher disappears.
Continuity mistake: Before the final board and zoning meeting finishes at the end of the episode, there are a number of people in the audience and a stenographer. As Hank is leaving they have all gone. It would have been near impossible for the entire room to vacate that quickly as less than half a minute had passed between seeing them and the room being completely empty.
Continuity mistake: When Luanne gets home from work at the start of the episode, Bobby is waiting for her to get the food she brings home. She puts the duck shaped bag on his tray with the beak of it facing Bobby. The next shot its facing the other way.
A Beer Can Named Desire - S4-E6
Continuity mistake: As Hank is discussing the competition with Peggy he is wearing his light blue boxer shorts. After Peggy asks whether he is going to still enter the competition, they are now dark blue.
De-Kahnstructing Henry - S3-E13
Deliberate mistake: In the scene where Hank is mowing his and Minh's (Khan's wife) lawn he mows straight across from Minh's lawn to his own lawn. This would be impossible as in every episode including this episode (before and after this scene) there is a fence separating his house and Khan's house. It was likely removed to simplify Hank's going back and forth between the two ladies as they try to gain his affection with food and beer. (00:15:30 - 00:16:40)
Peggy Hill: the Decline and Fall (2) - S4-E1
Continuity mistake: Hank discovers Peggy lying on her back in the field. In the previous shot Peggy fell face first towards the ground. There wasn't enough time for her to turn around before hitting the ground.
Deliberate mistake: As Dale is taking photos of Bobby's ventriloquist doll you can see he is taking the photo from the upper right of where Bobby is standing in his house. However the angle of which he takes the photo from would be impossible as it's to the left of their screen doors and there is no window or see-through material from that angle.
Continuity mistake: At the start of the episode when Hank gives Bobby the turkey knife it is in a red padding. Just as he's about to carve the bread (for practice) the padding is now yellow and stays that color for the rest of the scene.
Continuity mistake: During the scene where Bobby is accusing Hank for the first time of taking Native American land you can see Bill is standing behind Bobby. In a close up shot of Bobby, Bill is now standing to his right. As Bobby starts walking Bill is behind him again.
Continuity mistake: Whilst the old man in the retirement home is about to give Bobby the ventriloquist doll, there is a shot of just Bobby. In this shot you can see the jacket and hat on the coat rock and the coat is on the top rack. When we see both of them together the coat rack has another rack above the jacket now.
Answer: In addition to Brian Katcher's response: it's supposed to be a joke that Dale is always suspicious about conspiracy theories, mysteries, etc, but he is "clueless" about his own wife having an affair. Of course, as Brian wrote, many fans think that Dale is pretending to be clueless because he doesn't want Nancy to leave him.