When Ensign Gomez accidentally spills hot chocolate on Captain Picard, initially she hardly spills any liquid on his uniform (most of the hot chocolate appears to spill on the floor and on her). Yet, later, as Picard is walking down the hallway to the turbolift and encounters Q, almost the entire front of his uniform is soaked. There was not enough liquid spilled on him to make such a stain. We also see a larger stain on Gomez's uniform as Picard leaves and she turns toward Geordi.
After the Borg take the Enterprise's shields down with their shield-draining weapon, they fire one more time. Once again, it appears to hit the shields around the Enterprise, even though they're supposed to be down. The fact that this shot disabled the warp engines with no damage to the ship, indicates that those "shields" were actually the warp field. Numerous appearances of warp fields throughout the series show that they do have roughly the same ellipsoid shape as the shields. Although normally a warp field isn't supposed to glow under fire like the shields do, the Borg torpedo, unlike other energy weapons, was obviously designed to impact the warp field, thus creating a similar glowing effect.
As Guinan warns Picard to protect the Enterprise from the Borg, Picard orders the shields raised. Moments later, even though the shields are active, the Borg beam over. Considering that they are technologically much more advanced, it is conceivable that they may have found a way to transport through shields.
Data says that the damage done to a planet is identical to the destruction of the Federation outposts in Star Trek: The Next Generation: The Neutral Zone, implying that the Borg were responsible. This means that the Borg already know about the Federation, which seems to contradict Guinan's warning that the Borg will attack, now that they know about the Federation. Even if the outposts were too small to get the interest of the Borg, the Borg would have likely assimilated the people living at the outpost and thus their knowledge. Even if they didn't the outposts would have contained hints that they were part of a much larger and highly advanced civilization. Due to the fact that the Borg attacked both Romulan and Federation outposts they would also have learned that they were deal with two separate cultures on less than friendly terms with each other.
After the Borg "Slice the saucer section up like a roast", Picard orders Worf to "Use any force necessary...terminate that beam"...(unclear whether Picard meant the tractor beam or the "laser beam". However...Worf's first shot is absolutely nowhere near either beam, (it strikes low and right). Picard (seemingly oblivious by the inaccuracy) orders another shot. Worf's second shot is AGAIN low and to the right of both beams. Finally Worf's third shot hits the source of the tractor beam and immediately disables it. Doesn't this scenario seem improbable...that Worf (the HEAD of Security and Tactical)would fire the ship's weapons with such inaccuracy? (Probably similar to a fresh-academy Ensign's performance). There was no indication that Worf's targeting sensors were not functioning.
When Picard enters the turbolift after Gomez spilled her hot chocolate on Picard, the Turbolift doors mark he is on Deck 10, while the doors immediately to the right, denote his correct location of Deck 36.