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).
Other mistake: A final contest is given for control of "green stamps" with all the kids building a "House of Cards." Whomever causes the cards to fall will be the losers. All is going well until Tiger comes in and knocks into Greg and causes him to bump the table and knock over the cards. If you watch closely at the end of the scene, when all the dialog is finished, you can see Mike trying to knock down a few remaining cards. However they keep popping up. Probably due to the fact that the cards were glued together to make it easier for the kids to build upon.
Continuity mistake: When the kids are building the card castle to see who would get to choose what they get with the stamps, Jan is putting a card on, in one shot her hair is up in a ponytail then in the other shot her hair is down. This happens back and forth when the shots change.
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)
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."
Other mistake: At the end of the episode, when they were looking at the chart they created, and Alice was found to add the measles, you see Mike Brady adding the word "Alice." But the kids and Tiger all had columns, but he is adding her as an extra row, which was for the different illnesses, to suggest that Alice was an illness instead of a person having the illness.
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 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)
Visible crew/equipment: While "operation Alice" is underway, after Carol tells Alice to stop organizing the canned goods, when Alice places the step stool chair against the wall the black tape mark is visible on the floor where Carol had been standing, and then it's gone as the scene continues. (00:19:00)
Continuity mistake: When Carol and Mike are bringing stacked lunch trays upstairs to the kids, the milk glasses on the middle and bottom trays that Carol's holding switch to the opposite sides twice, in the wideshots and closeup. Then when Carol picks up the girls' stacked trays, the glass of OJ is on the bottom tray (after Peter's reaction to broccoli, it looks as if the OJ glass is gone), then as Carol walks into their room the OJ is now on the middle tray.