"That teacher is not fat."
You can, but I think you won't care about using a honorific word while gossiping somebody's weight.
The following is a direct quote from the module Greeting 2:
"In English, we say I am good or She is happy, but in Chinese you don’t use words like am or is before adjectives. Instead, adjectives are usually preceded by 很 (hěn). Sometimes 很 means very, but it’s more often just a way to connect a noun and an adjective."
Applying this principle to the translation of the sentence, "That teacher is not fat", I put "那个老师不很胖" and was marked wrong.
Am I incorrect or did Duolingo goof? If the former, why?
That would be "That teacher is not very fat". It implies that the teacher may be a bit fat, just not very fat.
But "hen" is often used as a neutral grammatical particle, even is this course, without meaning "very"... 他很好 doesn't have "very" here :-\
Every Chinese grammar I have had in my hands says it's grammaticalized. Notice how the Chinese always use the structure "very [adjective]" as a calque from their language. Some courses even consistently use 非常 as translation of "very".
I think I don't have more advice for you here. Anyway you can perhaps feel already such theory could not explain why you were wrong. So if you are already intermediate or advanced level and want to have an in-depth understanding it is advisable to look a bit outside of the box.
Today I think of an example. In English we say someone is healthy if he is not ill, but we can also say "Oh don't worry he is very healthy." Can someone be logically healthier than a healthy person? He has a negative number of illness? LOL. It is certainly reasonable to understand another language based on the reasoning of one's mother tongue, but we need to be aware that there is always limitation in doing so.