One used equivalence relation in regular languages is that for a language $L$ over alphabet $\Sigma$, $(x\sim^Ly)\Leftrightarrow (\forall w\in \Sigma^* xw\in L \Leftrightarrow yw\in L)$.
That means, if two words $x,y$ are not in the same class, there exists $w\in \Sigma^*$ such that $xw\in L$ but $yw \notin L$.
It is implied by Myhill–Nerode theorem, that if we have infinitely many equivalence classes for a language $L$, that $L$ is not a regular language.
My question is this: Is it correct to say that if we have an infinite amount of equivalence classes then there are infinitely many different $w\in \Sigma^*$ that "breaks" such equivalence relation between those classes? If so, how do we prove it? If not, is there a counter-example?