Plot hole: In the Upper Class Twit of the Year Show, there are five contestants. However, after Oliver runs himself over, in the events that follow there are only four props for the remaining twits (four mannequins, rabbits, and guns); since Oliver's death was unforeseen, shouldn't there be five of each? (This mistake, by the way, is rectified in the film version of this sketch).
Factual error: When the allies are testing their joke on the private in the field, just before he starts laughing at it, a bright red car drives past behind him. You can see it in a break in the trees. This sketch is set in WW2, and the car is obviously not from that era.
Audio problem: In the 'PC Pan-Am Gives Evidence' sequence (part of Series 3, Show 1), there is a loud, off-camera crash at 16'15", followed by hysterical audience laughter and a Palin visibly trying to stifle a laugh. This is because the heavily-bandaged 'Njorl', who had been replaced at that point by a dummy (enabling his head to be removed in the next scene), fell backwards off the podium, and had to be re-positioned by the crew. Filming presumably stopped, as a cutaway of Njorl appears in the middle of Pan-Am's address. The edit is also very audible, as Palin's speech suddenly plummets in volume.
Continuity mistake: At the start of the interview sketch, and at several times throughout, Cleese says he's interviewing people for a management training course. At the end, however, he suddenly claims it was a job interview.
Other mistake: William Knickers' letter, complaining about the sketches, starts "I strongly object ..." but the voiceover says "I object strongly ..."
Continuity mistake: At the beginning of the vegetarian restaurant sketch, Eric Idle stands right behind the pointy sculpture. It is quite short and in some shots the points of it are visible right at the bottom of the screen, but in other shots people stand and walk through right where it is meant to be. And in the final shot, when Eric is complaining about the punchline, it is definitely gone.
Deliberate mistake: Eric Idle's 'Stock Exchange Report' (in Series 3, Show 1) has a very obvious edit in the middle. This is a monologue delivered without cutaways, so the jump in the video tape is very easy to spot. Whether this edit indicates material removed or a simple joining together of two takes is unclear. However, since Idle gets drenched with water at the close of the routine (and given that, under BBC constraints, there was no time for drying out), we can assume that he must have made his fluff quite early on in the piece.