A system of sets forms an {\em m-fold covering} of a set X if every point
of X belongs to at least m of its members. A 1-fold covering is called a
{\em covering}. The problem of splitting multiple coverings into several
coverings was motivated by classical density estimates for {\em sphere
packings} as well as by the {\em planar sensor cover problem}. It has been the
prevailing conjecture for 35 years (settled in many special cases) that for
every plane convex body C, there exists a constant m=m(C) such that every
m-fold covering of the plane with translates of C splits into 2
coverings. In the present paper, it is proved that this conjecture is false for
the unit disk. The proof can be generalized to construct, for every m, an
unsplittable m-fold covering of the plane with translates of any open convex
body C which has a smooth boundary with everywhere {\em positive curvature}.
Somewhat surprisingly, {\em unbounded} open convex sets C do not misbehave,
they satisfy the conjecture: every 3-fold covering of any region of the plane
by translates of such a set C splits into two coverings. To establish this
result, we prove a general coloring theorem for hypergraphs of a special type:
{\em shift-chains}. We also show that there is a constant c>0 such that, for
any positive integer m, every m-fold covering of a region with unit disks
splits into two coverings, provided that every point is covered by {\em at
most} c2m/2 sets