Is There a Doctor in the House? - S1-E13
Plot hole: Lots of factors here: The doctors end up with the opposite kids that they were supposed to have... Why didn't they simply switch in the first place rather than go through all the drama? Also, the male doctor admits Alice let him in and most likely Alice would have told him first door on the right (like she's given directions in other episodes) to locate Peter. And the biggest question of all is, what doctor would make a house call and not look for their patient rather than just ending up with someone else they were not called to see (remember, each parent called their own doctor to see their biological kid).
Suggested correction: The lady doctor goes to see Peter as Mrs. Brady directs her to Peter's room first. Alice likely knows Peter already has a doctor looking after him, as it can be assumed Carol told her, so she sends the male doctor to Jan's room likely explaining to him that Mike's new daughter is also sick and needs to be checked. As to why they didn't solve this doctor problem to begin with; simply they are still a new family and didn't yet think about this as this is the first time the kids are sick.
Alice's September Song - S2-E23
Revealing mistake: One of the times that Sam comes over to visit Mr. Brady, he comes in through the back sliding "glass" door. There is no glass in the door. You can tell this because when Sam closes the door, to do so, he puts his hand through the areas where glass would be.
Is There a Doctor in the House? - S1-E13
Audio problem: When Mike walks into the kitchen and sees the chart with the list of kids on it and what ailments they've had, he comments on one of the ailments, saying "None of the kids have had distemper." However, if you watch his mouth, he appears to have originally said "rabies."
Continuity mistake: In episode 5-8 the girls have their walls wallpapered from pink to a yellow pattern with a white background. In the next episode it is back to the pink walls. The following episode it is back to the new wallpaper.
The Un-Underground Movie - S2-E4
Continuity mistake: During the filming of Greg's movie, Mike is in the stockade when Peter dumps the box of "snow". When the family sits to watch the film, Mike is released from the stockade by Carol before the snow is dumped. (00:17:45 - 00:20:10)
Continuity mistake: Cindy's notebooks keep changing in number and color during Peter's confrontation with Buddy. (00:21:30)
Continuity mistake: When Greg and Marcia are taping the fake message for Peter, Greg says, "Bang, he buckles down and brings home an A." But when Peter listens to the message a minute later (16:15), Greg's statement is now "Bang, he brings home an A." (00:14:50)
Continuity mistake: The shadow (supposedly of the window frame) on the headboard of Jan's bed changes from one shot to the next. (00:24:35)
Plot hole: When Mike was at the meeting and pulled out the Yogi Bear poster, he said he could find the other cylinder, and was told by one of the other men that they had to catch a plane in 30 minutes. After that, he had to find Jan, and then Jan and Marcia had time to go on about 4 rides and then find the sketches and then start their run across the park. There is no way all of that could have been done in less than 30 minutes.
Continuity mistake: When Marcia takes her last turn placing a card at the top of the tall house of cards, in the shots facing her she's stressed about accidentally collapsing the cards so she holds back her charm bracelet with her left hand, but in the first shot facing the three boys Marcia's left arm is down at her side and the charm bracelet is dangling at her wrist, right beside the cards. (00:21:00)
Everyone Can't Be George Washington - S4-E12
Continuity mistake: When Jan's in the backyard checking on the cloud and moon set decorations being painted by Marcia and Cindy, there's less "dark and gloomy" paint actually painted on the cloud after Cindy stands up with the smiley moon. (00:06:20)
Plot hole: At the end of the episode, the younger kids are singing the song "Girl" that Davy Jones did a little earlier in the show. The problem is that the song was just recorded and hasn't been released yet, they couldn't possibly know the song. Davy gives Marcia an LP, but it is on the same day the song was recorded, it couldn't be on there either.
Continuity mistake: In the photo of the football reception, the enlarged photo is not from the original. The receiver's legs are in a different position. (00:22:35 - 00:23:20)
Confessions, Confessions - S2-E12
Revealing mistake: At the start of dinner, in the wide shots we can see the table and chairs directly on the carpet. When Carol notices the leaky vase and calls for Alice, in the wide shot as Alice approaches the table, now there's plastic sheeting spread across the carpet under the table, as well some visible tubing under the table, presumably for the water stunt. (00:05:45 - 00:08:15)
The Impractical Joker - S2-E13
Continuity mistake: While Myron is on the closet shelf with Carol's shoes, the camel colored shoes repeatedly change between having camel colored heels to dark brown heels. (00:11:35)
The Voice of Christmas - S1-E12
Character mistake: Alice is in Mike/Carol's bedroom trying to find a place to hide a present and can't find room in any closets and chooses under the bed (only to find Bobby there first with the same idea). Why couldn't Alice simply hide the present in her own room?
Audio problem: When Marcia comes downstairs before the dance Alan shows her his new braces, and as they both head toward the door we hear the clear sound of shoes walking on a wood floor, even though they're walking on carpet. (00:23:50)
Visible crew/equipment: While Alice is hemming up Marcia's party dress, when Bobby and Jan walk into the family room the actor's black tape mark is visible on the floor in front of Bobby's feet. Note that it wasn't on the floor at the start of the scene. (00:10:35)
Visible crew/equipment: After Alice trips on the Chinese checkers, when Mike gathers the kids in the family room to update them, the shadow of the moving boom mic is visible at the top right corner of the screen. (00:02:50)
Kitty Karry-All is Missing - S1-E7
Visible crew/equipment: After Bobby walks out of the family room playing his kazoo, when Cindy places Kitty Karry-All on the couch so she can get the bottle, the actor's black tape mark is visible on the floor beside the couch, where Bobby had been standing. (00:02:40)