I came across 柄にもない once while reading and since the immediate look-up didn't feel like a full explanation of the phrase, I've been motivated to try and understand it connotations better.
In a quick jisho.org look up, 柄にもない means unlike oneself, uncharacteristic. Thesaurus lists it as 'words/actions which are not suitable for one's natural personality or impression', with alternatives such as 'not the type to do that' 'not that kind of character' etc. which seemed straightforward enough for me at first glance.
However, two things about that definition made me pause. First, the usage of ふさわしくない. 'Unsuitable, inappropriate' is used rather than 'dissimilar, contrastive'. This speaks to personal opinion having a play here rather than pure logical contrast. Second, the word 'impression' included along with personality, which, again, speaks to personal opinion rather than reality.
This made me start to think that 柄にもない wasn't so much 'against one's true self' as it was 'against the expectations of one's self'.
This opinion further strengthened when I looked it up on goo, where it's worded as 'something which isn't suitable for one's position, status or abilities, personality.' The part about not suiting one's status is a new usage, but again, it speaks more to expectations rather than intimate knowledge of a person.
While it may be a simple nuance, I think it really makes a difference in how one understands the term, especially while reading.
'Don't speak out of turn.' is the given translation on jisho and the impression of the wording, but that given translation doesn't fit into the literal meaning of 'don't say something unlike you'. It only works if you take '柄にもない' to have added connotations of expectations upon you. In this case, it isn't 'unlike you' but 'unsuitable for you [with your position etc.]'
There's no given translation for this on goo. I'd translate it as 'To say something uncharacteristically admirable', and I think it would be correct. But again, even this doesn't work if you read it with a nuance of being truly unlike yourself. If it's truly unlike you, then those admirable words are by default insincere, and likely lies. But, if it's just unlike what people expect of you, then the words are still sincere, if surprising and unexpected.
I think it's that word, the 'unexpected' nuance that 柄にもない speaks to the most.
For comparison, using らしくない for 'unlike you' is a wording that feels more intimate and accurate, as I feel like it implies the speaker really knows person. 'This isn't like you', 'it doesn't feel like a you-ish thing to do', 'it isn't something that seems like you'. The impression there is that the speaker has a deep and accurate knowledge of the person's character, and knows when something does not fit.
柄にもない, on the other hand, feels more like a general impression of going against a stereotype. It speaks more to expectations than knowledge. 'Not the type to do that', 'it's not their character', 'it doesn't suit your personality'. Like how you expect a Sergeant to speak according to their rank, you expect a dishevelled person to be irresponsible, you expect a studious person to be punctual, etc. The speaker is surprised the person is going against what they expect from them, rather than actually feeling like this goes against intimate knowledge of their character.
So yeah, that's my current impression, that the the meaning of 柄にもない doesn't seem to read as 'acting unlike oneself' so much as 'acting unlike the expectations of one's self'. But of course I could be wrong, which is why I'm turning to those more experienced and knowledgeable. I'd really appreciate other opinions from those who can confirm/joss the connotations, nuances, and usages for this word.