Summary: Three common errors that make narratives non-stories:
– Character does the right thing for the wrong reasons
– Character already knows the thing they “learned”
– Author passes the protagonist ball among characters for a single character arc
In 335 BCE, Aristotle kicked off literary theory by writing
Tragedy is a representation of a serious, complete action which has magnitude, in embellished speech, with each of its elements used separately in the various parts of the play and represented by people acting and not by narration, accomplishing by means of pity and terror the catharsis of such emotions.
(Did you notice the origin of “Show, don’t tell” is in that quote?)
Today we’d say that much of what Aristotle said about tragedy and comedy was, well, wrong. But it’s more useful than a lot of later theories, because Aristotle was also something of a logician and an ontologist, and tried to define drama in terms of necessary and sufficient conditions. 
The problem with later literary theories is that they were developed by non-logicians looking at famous literary works and trying to come up with things they all had in common. It’s like trying to define “bird” by looking at a hundred different birds, and coming up with, “X is a bird if X has wings.” You didn’t look at airplanes or bats or beetles, so you didn’t realize that having wings is necessary but not sufficient.
For instance, Aristotle says that a story’s protagonist must be virtuous, as people do not want a story about “villains making fortune from misery”. Later literary theories fail to mention this point! Every later theory I can think of would happily accept a story in which the protagonist was morally repugnant, and “grew” by “learning” a new way to exploit people. Hell, they even gave a Pulitzer to one such story, A Confederacy of Dunces. 
(Let’s get this clear up front: When I talk about whether something is a story, I don’t mean whether it satisfies anybody’s definition of a story. I mean whether it feels to me like a dramatic story. A Confederacy of Dunces does not feel to me like a dramatic story. But I’m not counting comedy in my definition of story. Like poems and songs, comedy is probably different enough to get its own rules.)
So it’s easy to write something that satisfies most theories about what a story is, and find when you’re done that it isn’t a story.
Here are some other common kinds of narratives that look like stories if you compare them to a “Hero’s Journey” or other checklist, but are not:
– Stories that wrap up the plot, but don’t then relate that plot resolution to larger character issues.
– Stories in which the theme is told rather than shown. Covered in “Show Us The Theme.”
– Stories in which the character does the right things for the wrong reasons. Take the scene in Return of the Jedi where (spoiler alert!) Darth Vader saves Luke from the Emperor. Suppose that what really happened is that Vader had just remembered that the Emperor was a big jerk and kept threatening to kill him, and this was a nice chance to off him and rule the galaxy. That wouldn’t be a story.
– Stories in which the character already knows the thing she’s supposed to be learning. Suppose Ebenezer Scrooge were a generous soul who loved Christmas at the start of “A Christmas Carol”. Not a story. This is okay if it’s a pure adventure story with no character growth, but it’s disastrous when the story is framed to look like a story about character growth.
– Stories which pass the protagonist ball for a single character arc. Suppose that in The Old Man and the Sea, the old man had hired a tourist to go out and catch a really big fish for him. And then suppose that the tourist tried and tried and was finally able to catch the fish because his fishing guide finally overcame his fear of shiny pointy things and attached a real fish-hook to the end of the line. One person had a problem; another tried to solve it; a third grew in a way that allowed it to be solved. Note that this is possible when you have multiple people in a group cooperating and therefore sharing “intentionality” (goals and intentions). But the tourist doesn’t know what the old man’s problem is, and never discovers that he’d been fishing without a hook all this time. The protagonist’s character arc is split across three people. No one person experiences the entire arc.
It also isn’t that cut and dry as you might be trying to match this story pattern:
– Character X has a problem.
– Character X tried but fails to solve the problem.
– Character X grows in a way that enables him/her to solve the problem.
You can fill in the blanks and say the story matches the pattern but it still might not be a story if you don’t be aware of the issues I pointed out.
The three biggest problems I’ve ever seen in indie writing (or even my own work for that matter) is:
-If the character already knows the thing they’re learning.
-Doing the right thing for the wrong reasons.
-Passing the protagonist ball.
These three problems appear as separate when we define a story in terms of a protagonist’s character arc. I think that if a story had all three of these problems it’d most likely be because they’re all manifestations of a single underlying problem: A story is (usually?) a moral lesson, and this narrative doesn’t have any moral lesson.
I don’t know if a story has to have a moral lesson, but most of the great ones have some kind of lesson.
For a better understanding of what is and is not a story, we’ll have to dig deeper than the formula of “character X grows to overcome a problem”, and probably talk about morality.
 Necessary and sufficient conditions are now known to be insufficient to define natural linguistic terms. This was discovered by philosophers in the 1930s, then by anthropologists and linguists in the 1960s, and then by artificial intelligence researchers in the 1990s, all independently of each other, because people are stupid that way.
 The specific point of the virtue of the protagonist may be overlooked because it isn’t fashionable anymore to reference morality in literary theory. If, as I argued in “Sex, violence, and meaning”, stories are morals, this causes problems for modern literary theory.