Character mistake: At the end of the episode, after Lore has been defeated, Picard asks Data if he is O.K. Data replies, "I'm fine." One of the plot points of this episode is that Data cannot use contractions.
Character mistake: In his exchange with Admiral McCoy, Data uses several contractions (i.e., shouldn't, I'm). For the rest of the series he doesn't, and in S3: Ep16, "The Offspring", he admits that his programming makes him incapable of using contractions. And in S4: Ep8, "Future Imperfect" Riker questions Data on when he started using contractions, knowing that Data is incapable of it.
Character mistake: When Data is on the planet, about to drop the material into the temporal rift, one of the three Datas asks which one of them should do it. The middle Data replies, "Me. It's me." Data is meant to be incapable of using a contraction.
Factual error: Picard reaches for the bowl of rotten fruit, winces in pain, and suddenly his fingernails have grown about an inch. This was due to the fruit bowl being inside an area where time was moving much faster. The problem is, in order for his fingernails to grow, blood would have to supply the needed nutrients to his fingers at the accelerated rate. Since his heart is in normal time, being away from the bowl of fruit, there is no way his fingernails could have grown like that - his heart is only supplying a normal-time-continuum's worth of blood.
Plot hole: During a private conversation with Worf, (the not yet revealed to be clone of) Kahless recounts a time during Worf's childhood when the actual Kahless appeared to Worf in a vision and told him he would do something no other Klingon had ever done. However, it would be impossible for a clone to have such a memory, as his creators would have no knowledge that such a memory even existed, let alone the circumstances and specific content of that vision.
Plot hole: They have already shown that anything that comes into contact with the time bubbles is immediately affeceted. The engine running 47 days non-stop, the fruit againg, Picard's hand aging, etc. Geordi creates subspace isolation field that allows them to beam off the shuttle and onto the other ships to investigate. Problem is, the second the transporter beam hit the time bubble, it would have stopped and never materialized because it would have been affected by the extremely slow passage of time in that bubble. The warp core breach was affected as was the disruptor fire so the transport beam would have been easily affected. And they have never shown that have been able to alter the transporters in such a manner that would allow them to beam over.
Stupidity: When Riker and Worf are searching for Geordi on the holodeck, why don't they just terminate the program, instead of looking for him in the simulated jungle? Would have made it a heck of a lot easier to find him in the relatively small empty holodeck, invisible or not.
Plot hole: The entire premise of the episode is contradictory; the franchise makes abundantly clear through the so called Prime Directive that the Federation abides to a code (often creating moral dilemmas that may require to stretch the rules) that says that their staff is not supposed to interfere in world that haven't reached warp capabilities, nor involve themselves in their internal matters. In the first half of the episode, Riker and the others just go 'mingle' with the frisky natives without a care in the world, and yet in the second half the Prime Directive itself is referenced explicitly and it is part of the plot.
Plot hole: The binars needed two people to upload the data back to their world but they only intended on keeping Riker in the holodeck. Riker is just one person. Picard came into the holodeck purely by accident - even Minuet said this was true. If Picard hadn't been there, Riker could not have uploaded the data on his own and the binars' plan would have failed.
Continuity mistake: In the opening log entry of the first episode, Picard says that "I have been informed that a highly experienced man, one Commander William Riker..." He sounds as though he has been assigned a First Officer without knowing him. However, in this episode he tells Pressman that he read Riker's file, and wanted him, based on Riker's adamant refusal to obey an order that went against his safety protocol.