We prove that for every
$k\,>\,1$, there exist
$k$-fold coverings of the plane (i) with strips, (ii) with axis-parallel rectangles, and (iii) with homothets of any fixed concave quadrilateral, that cannot be decomposed into two coverings. We also construct for every
$k\,>\,1$ a set of points
$P$ and a family of disks
$D$ in the plane, each containing at least
$k$ elements of
$P$, such that, no matter how we color the points of
$P$ with two colors, there exists a disk
$D\,\in \,D$ all of whose points are of the same color.