I've always had the impression that the Kobayashi Maru test was only given to Starfleet cadets. However, in Star Trek II: The Wrath of Khan we see Lt. Saavik (not a cadet, but a commissioned officer as far as I know) taking the test. Is there any explanation as to why an Academy graduate would take the test? Any level of canon is acceptable.
(My guess is it's essentially part of the bridge officer command test similar to what You takes in TNG)
From Memory Alpha:
In March of 2285, Saavik was a Starfleet cadet and, while mentored by Spock at Starfleet Academy, she underwent the Kobayashi Maru scenario. Despite an in-depth awareness of Starfleet regulations, she chose – during the test – to violate the Neutral Zone Treaty by venturing across the Klingon Neutral Zone in an attempt to rescue the freighter Kobayashi Maru, an action that resulted in the simulated deaths of all other officers in the bridge simulator. Afterwards, Saavik related to Admiral Kirk that she doubted the non-winnable simulation had been a fair test of her command abilities, though Saavik also conceded that she had not considered how possible such a situation was in reality.
Saavik was thereafter assigned aboard the Enterprise as navigator under Admiral Kirk, firstly in a training cruise that was then prematurely terminated so that the ship and crew could respond to the Genesis crisis. During these missions, Saavik was the highest-ranking cadet from a training crew on board the vessel. As such, she occasionally occupied the ship's command chair and was even permitted to supervise the vessel's departure from spacedock, which she had never done before.
This suggests that she was still technically an ungraduated cadet during these events, and her position aboard the Enterprise was part of a command-track placement, starting after taking the Kobayashi Maru.
It's also worth noting that cadets may take the Kobayashi Maru simulation multiple times if they so choose to (or are required to by tutors), and we do not know if this was Saavik's first attempt, or a later one.
Speculation: As Saavik was mentored by Spock, who in turn respected James T. Kirk, it is possible that Spock would encourage Saavik to take the test multiple times and see if the Vulcan would come up with a similar solution.
Saavik was, according to the film's official novelisation, a full lieutenant during the trip, but it's notably how young and inexperienced she (and all of the crew) are, this being a training cruise to allow young cadets and junior officers to learn ship's operations before being assigned to their real vessels.
As such, they are being relentlessly drilled and trained. Peter Preston is receiving maths lessons from Saavik in addition to engineering training from Scott, Saavik is getting command lessons from Spock, etc.
The final exam for the bridge trainees apparently involves the Kobiyashi Maru simulator. Saavik evidently hasn't seen this particular simulation program before, presumably because it's a command level test, not one that's normally administered (at least during this time period) to Academy cadets.
Captain's Log: Stardate 8130.5
Starship Enterprise on training mission to Gamma Hydra. Sector 14, coordinates 22/87/4. Approaching Neutral Zone, all systems functioning.
Mr. Spock, in his old place at the science officer's station, gazed around at the familiar bridge of the Enterprise. The trainees, one per station and each under the direction of an experienced crew member, were so far comporting themselves well.
It was a good group, and the most able of them was the young officer in the captain's seat. Spock expected considerable accomplishments from Saavik. She was young for her rank, and she enhanced her natural aptitude with an apparently inexhaustible capacity for hard work.
She is referred to as Lt. but I believe she is a Junior Grade Lt. The test at the beginning could’ve been a “Final” She was taking before receiving her graduating rank. It is uncommon but not unheard of for a cadet to graduate as a J.G. Lt Instead of ensign. She is shown to be an above average student so it would make sense that she could have been given the Junior Lt. rank upon graduation.
