The assumption is that normal speakers adhere to the maxims (or principle of Relevance, or whatever iteration of pragmatic theory you choose).
I once heard the argument that if someone does not follow at least the Cooperative Principle, they could be considered insane. That is, if one's goal of speaking is not to be understood and to understand others, then they are clearly outliers in society.
But different strategies for the maxims doesn't necessarily mean they aren't being Cooperative. They might just have a different perspective on what is optimal. This may vary between cultures. For example, politeness varies (see research on (Im)Politeness Theory) so that some cultures might be more direct. In Japanese (this is probably an oversimplification) a rejected invitation tends to be phrased as "maybe", which is understood as "no". That's a violation of a maxim (of quality), but it's a convention.
So when you find someone within your own speech community who violates the conventions for applying the maxims you expect, then you might find them to be outliers, or even insane, certainly difficult to talk to sometimes. (Note that "outlier" and "insane", etc., are all relative-- it could just as easily be you or I as the weird one.)
So to work on the phrasing here:
We can call the "maxims"
principles.
We can refer to Pragmatics as "conversational interaction".
And we can use the terms "norms" or "conventions".
Talking to someone is a lot like signing a contract: we agree to certain conventions of interaction in order to communicate effectively.
Someone who does not meet those conventions either isn't playing by the rules or lacks some sort of competence the others have, which we could call "following conversational interaction conventions" (or throw the word "principles" in there if you want).
One very interesting area of research is studies of how Autism Spectrum Disorder relates to pragmatic competence, or maybe better just to say pragmatic behavior. It's important to navigate the terminology carefully-- saying someone with autism is incomptent doesn't quite sound right, although certainly that phrasing has been used. The main point everyone can agree one is that there is variation in pragmatic styles.
To quote Sheldon from the
Big Bang Theory, some things are just "obligatory social conventions" and you have to do them, so he begrudgingly goes to his friend's birthday party or whatever. The same applies for conversations.
To directly answer the question: No, I don't know of a specific/official term that has been used to describe this as a "competence" per se, but the various phrasings above would be fine. This isn't specially my area of research (I do more with Semantics, and especially Syntax), but I'm familiar at least with what an introductory textbook would say and at least I don't think there's a widely used term outside of specialist circles. But there is a huge amount of research on pragmatics in interaction that I'm sure would be relevant to your questions.
More importantly, from the perspective of Pragmatics in general, the important part isn't whether you violate the maxims, but
why you violate them. If you violate them, it leads to a conversational implicature which also has communicative effect. We violate the maxims every day (maybe one could argue every sentence)! But understanding why and the inferences made when we do, that's what Pragmatics is all about. When in interaction the "why" is lost on the hearer, that's when the hearer would find the speaker to be potentially "incompetent". In other words, communication has failed because the conversational partners failed to understand each other's motivations. Think of it as this "why" question, and I think it will make more sense, and also be easier to explain. If you know why they did it, you understand their more general intent. If not, they just seem confusing. If there really is no "why" at all, then they might be insane. But if they have their own reason, just not clear to you, then you just have different perspectives-- from something as simple as not sharing background knowledge (maybe your friend thought it was obvious he'd use the WiFi at the train station, because he does that all the time, so it wasn't worth mentioning) to having very different perspectives about communication (such as whether or not it is important to be gentle and indirect in giving a rejection).
I would recommend looking at various examples of annotated Conversational Implicatures in context, along the lines of:
"It's cold in here."
Maxim of Relevance: why did he say that?
Interpretation: he must want me to close the window.
A very simple example, but you can see many more in an introductory textbook or on various websites.
https://en.wikipedia.org/wiki/Implicature#Conversational_implicatureAnd that's conveniently easy to explain, setting aside some of the terminology, to a more general audience. That can be your starting point.