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: Shawn fishes Mr. Feeny's watch from the septic tank and mentions the septic tank is deeper than it looked. Given this, he should be covered in human waste, instead of just looking a little dirty.