Picard orders the helmsman to proceed at warp 6, but, when he asks Data their ETA with the Phoenix, the latter says that they are traveling at warp 4. Considering the urgency of the situation the Enterprise would not have reduced speed. In addition Picard's order to intercept at warp 6 doesn't make sense either, a Nebula-class starship's maximum cruising speed is warp 9.5, very close to the maximum cruising speed of a Galaxy-class starship, which is warp 9.6. Maxwell knows the Enterprise is pursuing him, so it's highly unlikely he was traveling below warp 6.
While under attack from the Cardassian ship, Picard asks to increase power to the forward shields. In the next scene, the Cardassian ship is shown firing from behind.
When Captain Picard orders Worf to continue firing phasers, you hear the sound of photon torpedoes being fired instead.
In his opening log, Picard says that a long conflict between the Federation and Cardassians ended a year previously. It's odd that this conflict was never mentioned during the first two seasons, when it must have been going on.
Considering his unusual, and downright unstable behavior, it seems highly unlikely that the majority of Maxwell's crew would go along with what he was doing. His orders were in direct violation of Federation law, Starfleet regulations and the general principles of Starfleet. In that instance the Phoenix's executive officer, and the other senior officers, would not be obligated to follow Maxwell's orders, as they were unlawful. The very nature of his orders would obligate both the executive officer and chief medical officer to relieve Maxwell of command.
After the Phoenix has been located and Picard sends a warning to Captain Maxwell to break off his course of action, Picard tells Gull Macet that, "you see we are doing everything in our power to reach the Phoenix." Moments later, it is mentioned that the Enterprise is only moving at warp 4. The Enterprise has a top cruising speed well over warp 9. Had they been moving closer to that speed, they may have been in time to save the Cardassian ship.
When the Phoenix changes course to intercept the Cardassian supply ship Data reports it accelerates to warp 9. Picard orders the helmsmen to lay in a pursuit course and increase to warp 9 as well. However the Phoenix had a good 30 second head start on the Enterprise, in order for them to overtake it the Enterprise would need to be going faster than the Phoenix, Picard should have ordered the helm to go to warp 9.1.
When O'Brien volunteers to beam to the Phoenix even though its shields are up, he explains that the Phoenix uses a high energy sensor system that cycles every 5.5 minutes, with a window of a fiftieth of a second between cycles. He thus disclosed highly sensitive information regarding a vulnerability in the shields of the Phoenix's class of starship in the presence of a Cardassian officer. Given that O'Brien earlier showed his mistrust of Cardassians, he would have been less specific in his explanation in order to avoid disclosing such information to the Cardassian.
Another of "Chief" O'Brien's mystery ranks. Previously his rank was spoken as Chief Petty Officer. In this episode, we clearly see two solid gold pips on his collar, indicating a Lieutenant's rank. He is never referred to as anything but a CPO, but his collar jewelry keeps changing.
When chief O'Brien is at the bar in 10 forward the Cardassian orders a drink. He then asks if the chief wants another. O'Brien says he does and the Cardassian orders an "ale". Unless he's telepathic how does he know what the chief is drinking? It could be a an ale or a lager or a ginger ale or who knows what wacky space drink it could be.
Picard tells Maxwell that the only two options he has are for Maxwell to return to the Phoenix and command it back to Federation territory, or for the Enterprise to lock on with a tractor beam and tow it back in disgrace. This ignores the obvious option of removing Maxwell from command and letting his first officer (or a senior member of the Enterprise staff) take the ship back under its own power, which is indeed what happens at the story's end.
Early on, Picard asks the senior staff if "anyone else" aboard the Enterprise had served with Captain Maxwell, at which point he is informed that Chief O'Brien had served with him aboard the USS Rutledge. It was not yet established that anyone on the Enterprise had served with him, and O'Brien is the only one ever mentioned as having done so, so asking if "anyone else" served with him makes no sense.