A system of sets forms an 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 covering. The problem of splitting multiple coverings into several coverings was motivated by classical density estimates for sphere packings as well as by the 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 positive curvature. Somewhat surprisingly, 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: 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 at most c2(m/2) sets.