When Picard and Wesley are about to leave the Enterprise Wesley says: "Shuttle 02 ready for departure." but the one they are in is Shuttle 01. This is shown when the bay doors open and when the shuttle leaves.
When Picard met up with Wesley and Ensign Gomez en route to the shuttlebay, the turbolift doors show they are on Deck 10, whereas Shuttlebay 2 and 3 are on Deck 13.
Captain Picard is at Starbase 515 for an artificial heart transplant. The doctor begins by incising the abdomen, not the chest.
Commander Data has commanded the ship numerous times. He would know that the person in charge has an obligation to the ship over the captain. So why was he expressing concern that going to the ship in distress would be far away from where the captain would be? That makes no logical sense.
Picard describes a 2.4 percent mortality rate as "quite low." Yet compared to modern mortality rates for general surgery (1-2 percent), and assuming that mortality rates will go down with more advanced technology, a 2.4 percent mortality rate would be seen as alarmingly high by the 23rd century.
However, this is not general surgery: it is heart surgery, which has a higher mortality rate.
The shuttlecraft seen departing the Enterprise's bay and the shuttle en route to Starbase 515 clearly differ. The shuttle in the bay was boxier and the one en route was much rounder and streamlined.
During Picard's operation, the surgeon employs several laser scalpels. In a close-up, the scalpel is clearly wobbling, yet the laser beam remains steady.
When Riker demands to the Pakled captain that Geordi be released, the Pakled responds with "Request denied" and shoots Geordi with his phaser. But Geordi's arms flail outward and he's yanked back before the Pakled even shoots him.
At one point, a shot shows the dome above Riker's head on the bridge. The view appears to be dense with stars, although the shot just prior to this one shows Enterprise and the Pakled ship sitting in an area of space that contains far fewer stars.
When Geordi is rigging the Pakled's weapon system (around 38 mins), two units are seen label with English wording and which would have nothing to do with weapons, since they had labels for microphone connections, headphone connections volume controls, etc.
If a shuttle is incapable of warp speed and it is going to take six hours to get to Starbase 515, it makes no sense to send that slow vehicle on a trip that long when the Enterprise could whisk Picard and Crusher there in no time.
The Pakleds pointed Geordi's phaser at him, and demanded that he duplicate it in their replicator. Geordi could not have complied with their request, without them temporarily relinquishing control of the weapon. Since it was the only firearm they had, Geordi would have been in control, and able to return to the Enterprise.
Troi only informs Riker that the Pakleds are dangerous after Geordi transports over. Since this was the first encounter for the Enterprise, she should have been on the bridge from the beginning.
When Picard is telling the story of his youth, Wesley says that was right around the time the Klingons joined the Federation. The Klingon Empire has never been part of the United Federation of Planets.
Data tells Riker, "Our Betazoid counselor is often aware of things we are not." Considering Riker's familiarity with Deanna, which Data would be aware of, this seems an odd statement to make. (The line is also clumsy as Data mentions Deanna's race and position, known to the First Officer.)