Factual error: Shelby announces the Borg have entered Sector 001, and we get an exterior shot of the Borg ship flying past Saturn. The problem is that the Borg are approaching from the side of the planet illuminated by the sun, meaning they would actually be moving AWAY from Earth rather than toward it.
Continuity mistake: When Data and Troi are scanning the list of elements, searching for something to cause an explosion, the list is scrolling "up" the screen. When she tells Data to stop and go back, he does, but the list continues to scroll "up" instead of back "down".
Plot hole: When the image of the warp bubble closing in on the superimposed "Enterprise" image, Dr. Crusher asked for it to be the same scale for each image. At the rate the bubble was closing in on the engineering section, it would have been destroyed LONG before the 4 minutes 15 seconds the computer gave for it to be destroyed.
Continuity mistake: When commander Data is reciting the code to lock the rest of the crew out of the computer, he states the code '1-7-3-4-6-7-3-2-1-4-7-6-C-3-2-7-8-9-7-7-7-6-4-3-T-7-3-2-V-7-3-1-1-7-8-8-8-7-3-2-4-7-6-7-8-9-7-6-4-3-7-6-LOCK', but when the computer screen is shown the code reads '1-7-3-4-6-7-2-1-4-7-6-C-3-2-7-8-9-7-7-7-6-3-T-7-3-2-V-7-3-1-1-7-1-8-8-8-7-3-2-4-7-6-7-8-9-7-6-4-3-7-6-LOCK.'
Continuity mistake: At the end, Picard shows Data a baby born during the crisis. Neither of them touches it, but between shots, the baby's blanket changes positions. First it's almost completely covering his head and face, then it's pulled down and showing much more of him.
Continuity mistake: As the camera does a close up on the shuttle craft's forward window as it approaches the Enterprise, the dorsal section is seen getting larger the closer they get. As Data and Picard talk, the image fades out, and a minute later, the angle changes, the dorsal section is again seen getting bigger, and the ship disappears.
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Other mistake: Toward the end, Beverly gets in the turbo lift to go to Engineering. As it starts, you see the lights to her left strobe top down. This is to indicate movement. However, since the lift was at the bridge level, the lights should have gone from bottom up, showing that the lift is going down.
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Visible crew/equipment: In the scene in the cavern where Wesley fires his phaser to heat the rocks to keep him and Picard warm, Wesley's shadow is in the wrong place. The camera is in almost a direct line of sight between where the rocks are (which are supposed to be providing the light source) and Wesley with the cavern wall behind him. After Wesley fires his phaser and the room illuminates, his shadow should be either directly behind him or to the left of him, but his shadow appears to the right side of him. The only way his shadow would be there was if the light source was off camera a good distance to the left.
Plot hole: Why would the miners send a shuttle to get Captain Picard when the Enterprise has a compliment of shuttlecraft on board? Also, how could the Enterprise engineering team (led by Lt. Commander LaForge) miss that the thruster quad on the shuttle was near failure?