Data's Day - S4-E11
Revealing mistake: When Data asks the replicator for Feline supplement 74 and gives then gives it to Spot, a bug is crawling on the food. Replicators cannot create living organisms. (00:13:15)
The Arsenal of Freedom - S1-E21
Revealing mistake: As Data ducks to avoid the blast from the probe, the scene shifts to a wide shot. After Data tosses Yar to safety, he jumps after her. As he does, you can tell it's a stunt man with a bad wig.
The Arsenal of Freedom - S1-E21
Continuity mistake: After Geordi separates the ship, the saucer was well away from the star drive section, and they're at impulse power. When the star drive section goes to turn around, it is at warp speed, and it is a re-use of the high warp separation sequence from Episode One "Encounter at Farpoint."
All Good Things... (2) - S7-E26
Visible crew/equipment: Captain Picard joins the poker game and Data asks the Captain if he wants to deal. Picard says "Thank you Mr. Data." Riker sits down, and just before Captain Picard starts saying "Actually..." a camera or other bit of equipment briefly pops into shot behind Riker.
Other mistake: When the Pakleds shoot La Forge the second time, he raises his arms a bit, seemingly in preparation to go backwards.
Continuity mistake: As La Forge prepares to transport the Pakled shoots him. He reacts, and a second later reacts again.
Answer: Yes. The doors to the holodecks can be locked when in use by anyone aboard the Enterprise so they couldn't be disturbed. However, high ranking officers like Captain Picard could override the doors as it's seen that overrides are in use even for the crews quarters. Even Barclay, when he's indulging in one of his fantasies could have the doors to the holodeck unlocked by an override code.