Let $M$ be a compact, connected, orientable, hyperbolic $3$-manifold whose boundary is a torus and which contains an essential closed surface $S$. It is conjectured that $5$ is an upper bound for the distance between two slopes on $\partial M$ whose associated fillings are not hyperbolic manifolds. In this paper we verify the conjecture when the first Betti number of $M$ is at least $2$ by showing that given a pseudo-Anosov mapping class $f$ of a surface and an essential simple closed curve $\gamma$ in the surface, then $5$ is an upper bound for the diameter of the set of integers $n$ for which the composition of $f$ with the $n^{th}$ power of a Dehn twist along $\gamma$ is not pseudo-Anosov. This sharpens an inequality of Albert Fathi. For large manifolds $M$ of first Betti number $1$ we obtain partial results. Set \[{\mathcal C}(S) = \{\mbox{slopes } r \; | \; \mbox{ker}(\pi_1(S) \to \pi_1(M(r))) \ne \{1\}\}.\] A {\it singular slope} for $S$ is a slope $r_0 \in {\mathcal C}(S)$ such that any other slope in ${\mathcal C}(S)$ is at most distance $1$ from $r_0$. We prove that the distance between two exceptional filling slopes is at most $5$ if either (i) there is a closed essential surface $S$ in $M$ with ${\mathcal C}(S)$ finite, or (ii) there are singular slopes $r_1 \ne r_2$ for closed essential surfaces $S_1, S_2$ in $M$.