The question says all I need to know, but I will try to clarify it a little more.
Let $M$ be a compact 3-manifold with nonempty torus boundary such that ${\rm int}(M)$ admits a complete hyperbolic metric of finite volume. Is it true that there exists a closed 3-manifold $P$ and a link $\Gamma$ in $P$ such that $M$ is diffeomorphic to the link complement $P\setminus \nu(\Gamma)$? Here, $\nu(\Gamma)$ is an open, embedded tubular neighborhood of the link $\Gamma$.
It looks to me like it might be true, because you could fill the cusp ends of $M$ via Dehn filling, generating a closed hyperbolic 3-manifold $M_2$. Then, remove a "core curve" (and here is the problem, is this well defined?) of each torus which was a boundary of the original $M$ and now lives in $M_2$, and that would be your link. However, I didn't find the reference and would appreciate any clarification about it (including counterexamples or proofs assuming, for instance, orientability).
Thanks in advance.