Yes, there is a simple generalization of the Myhill-Nerode theorem that provides the characterization you seem to be looking for.
Let $\sim$ be the equivalence relation defined by the Myhill-Nerode theorem, i.e., $x \sim y$ if there is no suffix $z$ such that exactly one of $xz,yz$ are in $L$. Let $S$ denote the set of equivalence classes of $\sim$, i.e., $S = \{[x] : x \in \Sigma^*\}$, where $[x] = \{y \in \Sigma^* : x \sim y\}$. The Myhill-Nerode theorem tells us that any DFA for $L$ must have at least $|S|$ states. We can think of $S$ as the statespace of the minimal DFA that accepts $L$.
Now define $A$ to be the set of equivalence classes that contain at least one element of $L$, i.e., $A= \{[x] : x \in L\}$. Note that $A \subseteq S$. Also, we can think of $A$ as the set of accepting states for the minimal DFA that accepts $L$. In particular, any DFA that accepts $L$ must have at least $|A|$ accept states: consider any DFA $M$ that accepts $L$; then there is a function mapping the statespace of $M$ to $S$, and mapping the set of accepting states to $A$. (See the proof of the Myhill-Nerode theorem for the construction of this function.)
Therefore, $|A|=|\{[x] : x \in L\}|$ provides a lower bound on the number of accepting states of any DFA that accepts $L$, in terms of a property of $L$. If you find a language $L$ where $|A|>n$, then you know that there is no DFA that has $\le n$ accepting states and that accepts $L$.
