  Let $F$ be a closed essential surface in a hyperbolic 3-manifold $M$ with a
toroidal cusp $N$. The depth of $F$ in $N$ is the maximal distance from points
of $F$ in $N$ to the boundary of $N$. It will be shown that if $F$ is an
essential pleated surface which is not coannular to the boundary torus of $N$
then the depth of $F$ in $N$ is bounded above by a constant depending only on
the genus of $F$. The result is used to show that an immersed closed essential
surface in $M$ which is not coannular to the torus boundary components of $M$
will remain essential in the Dehn filling manifold $M(\gamma)$ after excluding
$C_g$ curves from each torus boundary component of $M$, where $C_g$ is a
constant depending only on the genus $g$ of the surface.
