The class schedules pretty much must be intended more as a plot device than any serious attempt at sitting down and figuring out how to schedule classes for four houses across seven years with only one teacher per subject. Any number of people have tried to figure this out, and I've seen a few claim to make work, though I do not have any of their results on hand. Still, a number of the problems should be self-evident if you pay any attention to detail at all. ChokolatteJedi wrote over 18 thousand words across 3 chapters trying to make sense of the schedules1 in what looks like a fairly well researched work that outlines most of the holes, problems, and inconsistencies.
I can accept that taking five electives on top of a course load that theoretically kept students well occupied for 2 years might over work and stress out even the best student. There might well be good reason for limiting most students to two of three electives. That being said, it boggles my mind that [Hermione] would be scheduled in three classes for first period Monday morning. One of two things is happening here. Either, as I said, the schedule is a total plot device and the three classes are on top of each other simply to make [Hermione]'s schedule obviously insane, or some of the classes actually have two sessions per year, and McGonagall has purposefully put [Hermione] in an "odd" session, perhaps attempting to make it less obvious that she is taking overlapping classes.
Of the two of these, the former, that it is purely a plot device actually makes more sense. Originally I argued this from a career requirements and Gryffindor class selection basis. It was a decent argument, but there is in fact a better that I had missed. [Hermione]'s exam schedule is the final bit of proof you need to demonstrate that third year Gryffindors are extremely limited without a time turner.
You can also make an argument from Bill and Percy's twelve OWLS.2 However, 1) we do not know that neither had a time turner, Percy is irritable enough to have had one; 2) you can argue that [Hermione], while very intelligent in some ways, shows incredible immaturity in others and that this may be the key difference between her lack of success at taking all the electives; and 3) we do not know that it is required to take the class to take the test. You may be able to sign up for OWLS in classes you did not take. The fact that [Hermione] does not do this is not proof positive that it is not allowed - she might well consider doing so "cheating" the way she considers the use of [Snape's][Snape] notes in the margins cheating in the Half-Blood Prince.3
[Hermione]'s schedule has a conflict between Charms and Ancient Runes exam times.4 We know from [Ron]'s inspection of her schedule on the first day of class that there she has overlapping times for Divination, Muggle Studies, and Arithmancy.5 Now, at least seven of the other Gryffindor students take Divination and Care of Magical Creatures. The latter is shared with the Slytherins, but the former appears to be dedicated to them. Either way, those two time slots have no conflict with the core classes everyone has to take.
Because none of the other seven named Gryffindor third years are known to take Muggle Studies, Arithmancy, or Ancient Runes, we are unsure if there is a normal Gryffindor time slot for them. We do not know what classes Sophie Roper or Lily Moon6 take as they are nearly entirely unmentioned in the books, but even if there was no interest (other than [Hermione]) that year, the time slots should be available because that should not be consistent from year to year. Still, you could argue that Hogwarts, for some truly bizarre reason, has no standard schedule and that the staff reinvents it each year. In such a case, with only one student interested, they may have … neglected to schedule some time slots. It is a viable argument, but ridiculous.
You could try to say that it has to be variable because interest is variable but while Divination seems to be a dedicated class, Care of Magical Creatures is not, and yet both have every single named Gryffindor in them. In fact, it is implied (but not stated) that most if not all Slytherin students took Care of Magical Creatures as well. So we have no problems with doubling up houses for the electives. My guess is that all electives are always doubled up, and that either we just do not hear about the other house in Divination, or that for whatever reason no one from that house is interested in Divination.
So assuming there is a normal time slot, there must be a way for a Gryffindor student to take Ancient Runes. The exam obviously cannot conflict with a core class everyone is required to take. [Hermione]'s exam does. The options are that the (again) that the schedule is a plot device, or that [Hermione] is in some other house's Ancient Runes class. Given my guess that all electives are doubled up, there are two Ancient Runes sections, one for Gryffindor and x, and the other for y and z. While the Gryffindor Ancient Runes roster is between one and three (depending on our two extra girls), we have no reason to think that the size of the class in x is equally small to justify dropping the time slot for lack of interest.
Since Divination, Muggle Studies and Arithmancy are all on top of each other, none of these three conflict with Ancient Runes. Thus there is no reason to shift her Ancient Runes class to another house's time slot, doing so would in fact make the situation worse, not better, given how tightly stretched the teachers are - the other house's time slot is almost certainly going to conflict with something else she is supposed to be in.
So the time turner is necessary because no student can take any two of Muggle Studies, Divination, and Arithmancy. This is confirmed because by dropping two of these, she no longer needs it. However, despite no longer needing it fourth year and beyond, she would still need it third year simply to attend both Charms and Ancient Runes, neither of which should have needed a time turner to allow her to attend the other three.
The excessive number of schedule conflicts are present because 1) they increase the mystery 2) it lets [Hermione] be super intelligent without informing [Harry] of anything 3) it provides reasons for her and [Ron] to fight.
[Harry]: </Harrypedia/people/Potter/Harry James/>/ [Hermione]: </Harrypedia/people/Granger/Hermione Jean/>/ [Ron]: </Harrypedia/people/Weasley/Ronald Bilius/>/ [Hogwarts]: ../ [Notes]: ../../
Footnotes
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ChokolatteJedi. Meta about Hogwarts Class Schedule https://archiveofourown.org/ Last Viewed 2020-06-29 ↩
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Mrs. J. K. Rowling. Harry Potter and the Chamber of Secrets p. 46; Location 607 of 4470. Pottermore Limited. American Kindle Edition. ↩
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Mrs. J. K. Rowling. Harry Potter and the Half-Blood Prince Pottermore Limited. American Kindle Edition. Exact citation needed. ↩
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Mrs. J. K. Rowling. Harry Potter and the Prisoner of Azkaban (p. 323). Pottermore Limited. American Kindle Edition. ↩
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Mrs. J. K. Rowling. Harry Potter and the Prisoner of Azkaban (pp. 101-102). Pottermore Limited. American Kindle Edition. ↩
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I know Fay Dunbar is the prototypical name for a Gryffindor girl. See my [Notes] on [Hogwarts] for why I am using these names. I pretty much have to pick between using a video game name or using a name from a note sheet. ↩