The phrase "mistake of destiny" implies that Mathilde believes that she has ended up in the wrong social class.

Believing herself to be of noble blood, despite there being no evidence to that effect whatsoever, Mathilde finds herself leading what to her in a humdrum lower-middle-class existence married to a humble civil servant.

As a consequence of her mistaken beliefs, Mathilde firmly believes that she is entitled to better things. So when her husband receives an invitation to the Education Ministry ball, Mathilde sees an opportunity to be the belle of the ball, to show off her beauty and rub shoulders with members of the social elite, to which she believes she belongs.

In order to do this, however, Mathilde has to look the part. Among other things, this means wearing an expensive necklace befitting of someone of her refined sensibilities. But because Mathilde really isn't anywhere near as refined as she thinks she is, because there was no mistake of destiny in her relatively low birth, she doesn't recognize that the supposedly valuable necklace she wears to the ball is in fact nothing more than a fake, rather like herself.