  We revisit the Maximum Node-Disjoint Paths problem, the natural optimization
version of Node-Disjoint Paths, where we are given a graph $G$, $k$ pairs of
vertices $(s_i, t_i)$ and an integer $\ell$, and are asked whether there exist
at least $\ell$ vertex-disjoint paths in $G$ whose endpoints are given pairs.
We present several results, with an emphasis towards FPT approximation.
  Our main positive contribution is to show that the problem's intractability
can be overcome using approximation and that for several of the structural
parameters for which the problem is hard, most notably tree-depth, it admits an
efficient FPT approximation scheme, returning a $(1-\varepsilon)$-approximate
solution in time $f(td,\varepsilon)n^{O(1)}$. We manage to obtain these results
by comprehensively mapping out the structural parameters for which the problem
is FPT if $\ell$ is also a parameter, hence showing that understanding $\ell$
as a parameter is key to the problem's approximability. This, in turn, is a
problem we are able to solve via a surprisingly simple color-coding algorithm,
which relies on identifying an insightful problem-specific variant of the
natural parameter, namely the number of vertices used in the solution.
  A natural question is whether the FPT approximation algorithm we devised for
tree-depth can be extended to pathwidth. We resolve this negatively, showing
that under the Parameterized Inapproximability Hypothesis no FPT approximation
scheme for this parameter is possible, even in time
$f(pw,\varepsilon)n^{g(\varepsilon)}$, thus precisely determining the parameter
border where the problem transitions from ``hard but approximable'' to
``inapproximable''.
  Lastly, we strengthen existing lower bounds by replacing W[1]-hardness by
XNLP-completeness for parameter pathwidth, and improving the $n^{o(\sqrt{td})}$
ETH-based lower bound for tree-depth to $n^{o(td)}$.
