If you don’t have a good story, you will not have a successful short animation.
Of course, “good” could mean a lot of things. In the context of animation production, it means, specifically, both “engaging” and “producible.”
It’s not a shocker to hear that a story must be engaging, but the need is magnified when you are dealing with the limited run length of a short project. With only a couple of minutes of the viewers’ attention available, you have to grab them right away, make your point quickly, and not stray from the theme.
That said, a weak story will not kill your production.