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.
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: 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.