Deliberate mistake: The dying Vantika hides his consciousness away in a portion of Dr. Bashir's mind. At one point, "Bashir" (heavily cloaked in clothing and shadows) comes up behind Quark and makes sure his demands are clear. At this point, it should still be unclear just how Vantika is still "alive", and whose body he is using. However, when he shoves Quark away, for a fraction of a second, Bashir's face can be seen under his hood. They were true to plot in using the right actor (though he shouldn't have been seen), it is possible some might have spied the secret earlier than the show meant to reveal it.
Deliberate mistake: In fight scenes, the Yellow Ranger's stature matches that of a male (and not Trini). This is because in the Japanese version of the show, the Yellow Ranger was played by a man.
One Breath - S2-E8
Deliberate mistake: When Scully is shown on her hospital bed, it's obvious she's wearing a bra under her hospital gown. It's doubtful that a patient in a coma would be wearing a bra as it could restrict her circulation.
Deliberate mistake: About halfway through the show, Roz and Daphne are in the kitchen talking. Daphne pours water into the teapot - but there is no steam coming from the kettle, nor any sound of a boiling/just boiled kettle. No self-respecting Brit would make tea with water that wasn't boiling.
Deliberate mistake: "Treasure of the Ages": Darien delivers Mary's child during the hurricane. There's no time lapse in the scene, but somehow the kid emerges perfectly clean: no blood, no placenta - and no umbilical cord. Darien also neglects to wash his hands before reaching in to turn the breeched baby around. (Ewwwwww!) (00:45:00)
Deliberate mistake: Morgan turns on the stereo and starts dancing on the sofa with her doll. She then tosses her doll, which tips over a plant and breaks the glass on a set of French doors. The stereo magically stops the moment the glass breaks.