Continuity mistake: Hank says that when he and Peggy were 18, they were already married. But in the season one episode "Square Peg", Peggy tells Luanne that she "never even kissed" a boy until she was 20.
Continuity mistake: When Bobby is trying to convince Hank's friends to join the abstinence club, and when he asks Boomhauer he just stays silent. Bill then signs up and gets one of the cards Bobby has. As Hank starts to talk it goes to a close up of him but in the next shot Boomhauer is looking at one of the cards. Bill still had his card so where did Boomhauer get one of those cards?
Continuity mistake: As Luanne is becoming a "Born again" virgin, she yells after being dunked in the water. She yells "I'm a Virgin." and there is a shot to the left of two guys in the distance playing Frisbee look behind to their left. The shot sees them from the left meaning they were to Luanne's left. A few shots later Luanne is getting out of the lake and one of the guys who was playing Frisbee comes up to her from her right side and the other guy follows him. There wouldn't have been enough time in the scene for them to get all the way over to the right side of the field and not much reason to either.
Continuity mistake: In the episode where porn videos are being sent to Hank, there is a close-up shot of a machine printing out address labels for Hank's house. These labels say 320 Rainey Street. However, in this episode, when Peggy is looking at the mail, all of their mail has 84 Rainey Street as their address. In several other episodes, the address is stated to be 123 Rainey Street.
Continuity mistake: After the frantic phone call from the hospital, Hank rushes to the emergency room. In the hallway there are two doors, the one to his left says "Consultation" whilst the one to his right says "Examination". He uses his left hand to open the door and in the next shot you can see the door now says "Examination" and "Consultation" on the opposite door.
Continuity mistake: When we see Bobby talking to Hank about the allergy test there are lots of black marks (dots for the allergy test) on Bobby's back and one red circle around one of them. As Hank is looking at them there is a shot from behind Bobby and the red circle has gone.
Continuity mistake: When Peggy says that Bobby could have invited her and Hank to the dog house, Hank says that they wouldn't have been able to fit in the door, but he went in and out of it earlier while fixing up the doghouse. At the end of the episode, Peggy is inside the doghouse with a student she is tutoring.
It's Not Easy Being Green - S5-E16
Factual error: Boomhauer's car, Ms. Sally, has a Texas flag license plate on it. These plates were not issued until 1992, but Ms. Sally was driven into the quarry in the late 70's or early 80's.
It's Not Easy Being Green - S5-E16
Continuity mistake: When Hank goes to Bobby's class for "environmental court" and is sworn in, you see Bobby sitting at a large desk and to the right there is nothing next to it. The next shot there are now a few empty desks next to the large desk.
It's Not Easy Being Green - S5-E16
Continuity mistake: As Hank tells Bill & Dale that they are going to drain the quarry (whilst in Dale's basement) and Dale lets his burning cigarette drop. Dale & Bill start to freak out and it goes to a wide shot of the three of them & the cigarette Dale dropped is not on the ground anymore.
It's Not Easy Being Green - S5-E16
Continuity mistake: When Hank, Dale & Bill go to the quarry to get the car, they use a tow truck to try and pull it up. If you look at the tow truck you can see it is a red color. Later on when the protesters come and are sitting in a circle and singing the tow truck is now orange. It stays orange for the rest of the scene.
Plot hole: As Dale comes to Hank's work and talks to Hank about the gun incident he has a spray pack on his back that is filled with poison and by accident sprays Joe Jack. Yet in the episode "The Exterminator" Dale was told by the doctors that he would not be able to use any type of hazardous chemicals with bug spraying as his lungs wouldn't be able to handle the fumes any more. Even if he ignored the advice the chemicals would have at least incapacitated him or killed him.
Character mistake: When Peggy and Lupe return to Mexico and Peggy thinks the crowd of people are cheering her on, she says "Por los niños!", pronouncing the "ñ" correctly. However, in several other episodes of the show, she pronounces the "ñ" in the word "español" incorrectly.
Other mistake: The premise of this episode makes no sense. Bobby and Connie have previously spent a lot of time together and/or with Joseph. They were also shown on "dates", i.e. the high school dance where No Doubt played. Basically, they were a steady junior high/middle school couple. However, in this episode, they are suddenly "too different" and can barely get through a few dates together.
Deliberate mistake: The Olympic symbol (of 5 inter-connected rings) is upside-down on the tracksuits the torch carriers are wearing near the end of the episode. It has 3 rings at the top, not 2.
Deliberate mistake: After Peggy has knocked King Philip off his horse at the end you can the sun is setting behind her, when it goes to a reaction shot of Bobby and Luann in the stands, it is a blue sky and sunny behind them. The following shot of Peggy it goes back to the sun setting. The mistake could be said to be deliberate as the shot was used earlier in the scene and probably used again to save time and money than re-animating the exact same shot.
Continuity mistake: After Peggy throws the tomato at King Philip you see it lands on the badge on his jacket. Most of the tomato lands about the top of his badge and seems to set on the top part. A few scenes later when Peggy is being put in the stocks for punishment you see Philip again and the stain has moved down the badge. Most notably the tomato stain looks larger and a large chunk of it has moved below the badge onto the shirt.
Continuity mistake: When Hank and the family are approaching Cotton's table at the rummage sale, there are two empty knife sheathes and one sheathed knife on his table. In the next shot, there are 3 sheathes and 2 knives out of the sheathes. (00:03:25)
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)
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.