The Enterprise arrives at Relva VII so that Wesley Crusher can attend the Starfleet Academy entrance examination. He runs into his friend Jake Kurland, a distraught young man who, unlike Wesley, has been turned down for the entrance examination. Despite some encouragement, Kurland still feels as if he has disappointed everyone. At the same time, Captain Picard is visited by Admiral Gregory Quinn, a Starfleet Admiral and close personal friend of Picard's. Quinn has arrived with Lieutenant Commander Dexter Remmick, a Starfleet Inspector who has come aboard to gauge the Enterprise for faults in crew performance and protocol. Picard demands an explanation, but Quinn maintains that this is a classified investigation that demands full cooperation.
Meanwhile, Wesley Crusher encounters his competition at the testing center. He will be up against T'Shanik, a Vulcan female, Oliana Mirren, a human female, and Mordock, a highly talented Benzite with superior mathematical skills. Starfleet Officer Chang arrives to congratulate the applicants on their highly esteemed talents, but reveals that only one applicant will be chosen out of the four for acceptance into the Academy. Without further ado, he begins the process of testing.
On the Enterprise, Commander Remmick begins his investigation, breathing down the necks of bridge crew members and thoroughly irritating Commander Riker. He confronts Picard in his ready room as to the real reason behind Remmick's visit. Picard shares his equal frustration with Riker, assuring him that he is also being kept in the dark as to the clandestine nature of the examination. Riker's temper begins to fuel when Remmick privately confronts him about discrepancies in Captain Picard's log entries, to which the Commander demands that Picard be present to answer for. Remmick refuses, threatening Riker to stay on course and do as he is ordered. He also confronts Deanna Troi, posing questions to her about Captain Picard's psychological stability and possible lapses in judgment, as well as Geordi La Forge about the actions of Mr. Kozinski which ultimately led to the Traveler accidentally flinging the Enterprise light years across the universe.
While on break, Wesley finds privacy in an empty holodeck. Lieutenant Worf approaches to ask him about his testing. Wesley admits his trepidation at having to face the final phase of his testing, which is nicknamed the "psych test." Worf reveals that Starfleet is exceedingly efficient at determining a candidate's worst fears and how they react to them, leaving Wesley amazed at how a Klingon officer could possibly fear anything. Worf boldly reassures Wesley that only "fools" have no fear.
Things become tense on the bridge when young Jake Kurland steals an Enterprise shuttlecraft for the purpose of running away. When Picard contacts the shuttlecraft, Jake Kurland tells of his shame at having failed to gain entrance to the Starfleet exam, and his inability to face his father because of that shame. The shuttle ends up losing power, heading straight for the atmosphere of the planet, putting Kurland's life in peril. With no possibility of a transporter lock, Picard realizes that Kurland must manually regain control of the shuttle before he burns up in the atmosphere. Picard authoritatively orders Kurland to do exactly as he says, issuing a command that Kurland angle the shuttle nose-first towards the planet, then force a restart of the shuttle's engines. Kurland hesitates, but eventually puts his trust in Picard and follows orders, causing the shuttle to bounce harmlessly off of the atmosphere and away from the planet. Remmick expresses severe disappointment, suggesting a severe lack of discipline on the Enterprise. Picard reminds the Commander that young men sometimes make rash and poor decisions, and should be reprimanded with proper discipline under an authority figure.
Meanwhile, Wesley successfully defuses a tense situation when another officer bumps into him in a corridor, prompting a heated, angry threat. At first, Wesley is diplomatic and apologetic, until he spots the officer's webbed fingers, a trademark of the Zaldans who view politeness and courtesy as an irritating form of dishonest social behavior. Wesley immediately threatens the larger officer with violence if he does not back down, something the Zaldan admires before congratulating him and walking away. Officer Chang approaches, having watched the confrontation. He commends Wesley on passing yet another test, the ability to deal with different races and their respective cultures.
Commander Remmick becomes increasingly frustrated at his inability to find any wrongdoing on the Enterprise. He confronts Commander Data on the issue of log tampering, to which Data calmly suggests an error in Remmick's judgment. He then approaches Beverly Crusher about her relationship with the Captain, a man who he suggests was responsible for the death of her husband. Dr. Crusher offers no personal details, remaining professional. Finally, he confronts Picard on the issue of violating the Prime Directive with the Edo species. Picard refuses to argue, standing by his log entries while suggesting that Remmick's search to uncover any wrongdoing to be a futile effort. He then confronts Admiral Quinn in his quarters, demanding an end to the inspection which has put the Enterprise crew on edge, and personally strained their friendship with one another. Quinn expresses his understanding, assuring Picard that it will soon be over.
Wesley and the other students engage in a semi-final test where speed and accuracy are key. Mordock finds difficulty with the complex test, at which point Wesley kindly offers assistance despite the competitive purpose of the testing phase. Mordock eventually gains the upper hand and finishes with an impressive score. Officer Chang informs the applicants that the final test will take place soon. Wesley questions Chang as to whether this will be the fabled "psych test" that he has been dreading. Chang corrects Wesley as to the true description of the test and its purpose, and offers his encouragement.
Commander Remmick delivers his report to Admiral Quinn with Picard silently standing by. Remmick reports that he could find no evidence of wrongdoing on board the Enterprise, adding his admiration for the distinction of the ship's crew. Dismissed, Remmick turns to leave, but not before expressing his wish to Captain Picard that he may someday be able to serve aboard the Enterprise. Picard, clearly annoyed, ignores the Commander and says nothing in response. When Remmick leaves, Quinn begins to explain the reason for the mysterious investigation. He expresses his concern over certain problems which he has noticed in the Federation of late. Despite the ambiguity of the claims, Quinn is nevertheless compelled to tell Picard that he believes some unknown force is attempting to destroy everything the Federation has worked so hard to achieve in the last 200 years. Even more startling is that Quinn believes the threat could very well be from within the Federation's highest levels of command. To properly gauge the situation, Quinn needs officers he can trust. He then tells Picard that he would like to promote him immediately to Admiral, and give him full command of Starfleet Academy. In such a position, Picard would be close to Quinn at all times in a key position. Picard demands time to think about the proposition.
Meanwhile, Wesley nervously awaits the psych test. He encounters Mordock who is in profound emotional duress following his psych test, but encourages Wesley that everything will be alright. Wesley is led inside a small room with empty chairs, ready for the test. Moments later, he hears an explosion outside of the room and heads out to investigate. He enters an environmental maintenance lab where a pressure tank has exploded, filling the area with hydrogen gas. The computer issues an evacuation warning that the doors will be sealed in 45 seconds. Wesley spots two officers; one trapped underneath a fallen beam, and the other cowering in the corner, unable to overcome his terror. Wesley helps the trapped and injured crew member, but is unable to talk the other one out of his fear. With no possible alternative, Wesley leaves the room, saving the injured crew member while sacrificing the other to his doom.
Officer Chang approaches to calm Wesley down in the hallway. In truth, the psych test has just taken place. Wesley is stunned by the revelation, even more so when the other crew member emerges unharmed from the environmental lab, smiling. Chang explains that Wesley successfully passed the test by making a decision to save one man while sacrificing the other, instead of trying to save both and dooming all three. The decision was neither right, nor wrong, but Starfleet officers are always expected to make such a decision in times of crisis. Wesley realizes that Starfleet correctly identified his greatest psychological obstacle; the decision that Captain Picard had to make when Wesley's father was killed. Later, Chang addresses the anxious candidates as to who will enter Starfleet Academy. Mordock is congratulated as the first Benzite to enter Starfleet Academy, something which disappoints the other applicants. Mordock expresses guilt over his success, telling Officer Chang that Wesley deserves the honor for helping him through the speed test. Chang assures Mordock that many factors were taken into consideration, and Wesley did not lose out because of that test. In truth, Wesley was simply not ready.
Captain Picard speaks with Riker about Quinn's decision to promote him to Admiral. Riker congratulates the Captain at such an esteemed new position. Nevertheless, Picard decides to refuse the promotion, leaving to tell Admiral Quinn at his farewell dinner. On the way he inadvertently bumps into Jake Kurland, addressing the young man and suggesting that running away is never a solution to any problem. He locates Wesley Crusher who is in the observation lounge, feeling ashamed at having failed the entrance exam. When he suggests that he has let down the Enterprise, Picard scoffs at the notion and says that he did his best, and will definitely do better next year when he re-applies. In an effort to boost the young man's confidence, Picard reluctantly reveals that he too failed his first entrance exam, immediately demanding that Wesley not tell a soul. A shocked Wesley finally realizes that even Starfleet's most respected officers sometimes fail, but do not let failure defeat them entirely. The two leave for the Admiral's farewell dinner.
Later, Picard and Quinn head to the transporter room for the Admiral's departure. Quinn expresses his disappointment at Picard's decision to turn down the promotion, but Picard assures him that he would best serve him as Captain of the Enterprise. Quinn agrees, departing with a warm "Safe travels, my friend."