Plot hole: About two thirds of the way through this episode, Data deduces how Moriarty was able to 'leave' the holodeck. The big reveal is that he never did leave the holodeck, he merely reprogrammed it to simulate the rest of the ship without Picard, Barclay or Data's knowledge. While it is believable that this would fool humans like Picard and Barclay, it is ludicrous to suggest that Data would be taken in by it, even for a second. Data is an android whose perceptions of sight, sound and the world around him are far more sophisticated than humans. To list what we know of Data's perceptive abilities from previous episodes would take up the entire page, but suffice it to say he should have immediately recognised the 'Enterprise' as force fields and holograms rather than the genuine article. Note: this goes beyond a 'character mistake' or anything like that. Data's enthusiasm for Sherlock Holmes style deduction should have led him to eliminate the impossible before considering what was probable. Data accepts the impossibility of a holodeck character existing is the real world before discounting the possibility that it was still a holodeck simulation. This contradicts an awful lot of what is known about Data's abilities and powers of deduction. Although it serves for a good mystery the fact is it takes Data far longer than it should to deduce, or even guess at, the truth of their situation.
Plot hole: When Picard, Geordi, and Troi encounter one of the first time disturbances it results in one of the runabout's nacelles using up all of its fuel. Data says this is due to it having been in operation for 47 days (according to the plasma conversion sensor). After this the Captain reaches for the bowl of rotting fruit which causes him to scream in pain. The other crew members rush in and Troi scans his hand. She tells Picard that his hand is metabolizing at approximately 50 times the normal rate. Data and Geordi then discover that the temporal disturbance which covers the fruit also covers the nacelle that has lost all of its fuel. Data also notes the disturbance extends outward from the hull, about 17 meters from the ship and is spherical in shape. The problem here is that according to what Data said earlier the engine was active for 47 days even though it had only very briefly come into contact with the time distortion - maybe 1 second at most, but likely far less time than this since the ship was at warp when the fuel was consumed. So if we assume the engine was in contact with the fragment for 1 second then time is actually moving at 4,060,800 times the normal rate - not 47 or 50 times normal. (00:11:00 - 00:12:00)
Visible crew/equipment: During the scan of the temporal rift at the beginning of the episode, the tip of the microphone briefly comes into view at the top of the screen above Captain Picard's head when Data says, "Nothing I have seen before." [Not visible in the remastered Blu-ray version].
Character mistake: Captain Picard wonders why the boy would want to go back to his captors if they abuse him and Dr. Crusher tells him it's not uncommon, it's called the Stockholm Syndrome as if Picard doesn't know or understand. However, in S03E12, "The High Ground", Dr. Crusher is held captive and when Picard is talking to her (after he is captured trying to rescue her), she begins to appear sympathetic to her captors and Picard says "I don't have to remind you of the psychological impact of being a hostage." Picard would be aware of why the boy may wish to return.
Other mistake: When Picard first meets Data in the past, Data's rank insignia is wrong. He's always held the rank of Lieutenant Commander, signified by two gold pips, and one with black in the center. However, Data has one gold, one with black, signifying Lieutenant.
Other mistake: As the Enterprise is fleeing from Q, it peaks around Warp 9.6 which according to the fact books, translates to around 1000c in sidereal space. Also according to the fact books, the saucer section is not capable of sustained Warp flight on its own. Also, at the beginning of The Chase scene, the Enterprise is on course to Farpoint, Q's fence is perpendicular to it. The Enterprise does at least a 90°+ turn beginning her escape maneuver prior to separation, so it is definitely not flying past Farpoint during The Chase. Still the saucer section somehow makes it to Farpoint a mere 51 minutes after the engine section. There's no evidence at all in the dialog that it was Q's doing.