Length of distinguishing string for a DFA
Distinguishing strings can, in general principle, be of any length, though the specifics will depend on the automaton and states that you're looking at.

For example, consider the usual automaton that accepts even-length strings over the alphabet $\{1\}$. We have states $q_\mathrm{e}$ (which is the start state and is accepting) and&nbsp;$q_\mathrm{o}$ (which is not accepting), and the transition function just swaps state every time a character is read.  For this automaton, every string over $\{1\}^*$, including the empty string, distinguishes the two states. Strings of even length (including&nbsp;$\epsilon$) distinguish because they're accepted by&nbsp;$q_\mathrm{e}$ and rejected by&nbsp;$q_\mathrm{o}$; odd-length strings are the other way around.