  In the \emph {barrier resilience} problem (introduced by Kumar {\em et al.},
Wireless Networks 2007), we are given a collection of regions of the plane,
acting as obstacles, and we would like to remove the minimum number of regions
so that two fixed points can be connected without crossing any region. In this
paper, we show that the problem is NP-hard when the collection only contains
fat regions with bounded ply $\Delta$ (even when they are axis-aligned
rectangles of aspect ratio $1 : (1 + \varepsilon)$). We also show that the
problem is fixed-parameter tractable (FPT) for unit disks and for
similarly-sized $\beta$-fat regions with bounded ply $\Delta$ and $O(1)$
pairwise boundary intersections. Furthermore, we can use our FPT algorithm to
construct an $(1+\varepsilon)$-approximation algorithm that runs in
$O(2^{f(\Delta, \varepsilon,\beta)}n^5)$ time, where $f\in
O(\frac{\Delta^4\beta^8}{\varepsilon^4}\log(\beta\Delta/\varepsilon))$.
