Dungeon Master Guide calls this Degrees of Failure or Success at a Cost
The opening paragraph of the Resolution and Consequences section of the DMG (p. 243) encourages non binary results to player actions:
As a DM, you have a variety of flourishes and approaches you can take when adjudicating success and failure to make things a little less black-and-white.
It goes on to addresses variable skill check results directly.
Success at a Cost
When a character fails a roll by only 1 or 2, you can allow the character to succeed at the cost of a complication or hindrance.
Degrees of Failure
Sometimes a failed ability check has different consequences depending on the degree of failure. For example, a character who fails to disarm a trapped chest might accidentally spring the trap if the check fails by 5 or more, whereas a lesser failure means that the trap wasn't triggered during the botched disarm attempt...
Varied Success by Skill Check
Also addressed is tiers of success based on the skill check in the social interactions section.
A neatly formatted explicit example is on DMG page 245 detailing reactions of NPCs based on the roll of the skill check.
\begin{array}{|l|l|}
\hline
\textbf{DC} & \textbf{Reaction} \\ \hline
0 & \text{The creature offers no help but does no harm.} \\ \hline
10 & \text{The creature does as asked as long as no risks or sacrifices are involved.}\\ \hline
20 & \text{The creature accepts a minor risk or sacrifice to do as asked.}\\ \hline
\end{array}