An incompressible bounded surface $F$ on the boundary of a compact, connected, orientable 3-manifold $M$ is arc-extendible if there is a properly embedded arc $\gamma$ on $\partial M - {\text{Int}} F$ such that $F \cup N(\gamma)$ is incompressible, where $N(\gamma)$ is a regular neighborhood of $\gamma$ in $\partial M$. Suppose for simplicity that $M$ is irreducible and $F$ has no disk components. If $M$ is a product $F\times I$, or if $\partial M - F$ is a set of annuli, then clearly $F$ is not arc-extendible. The main theorem of this paper shows that these are the only obstructions for $F$ to be arc-extendible.