We provide a degree condition on a regular n-vertex graph G which ensures the existence of a near optimal packing of any family H of bounded degree n-vertex k-chromatic separable graphs into G. In general, this degree condition is best possible. Here a graph is separable if it has a sublinear separator whose removal results in a set of components of sublinear size. Equivalently, the separability condition can be replaced by that of having small bandwidth. Thus our result can be viewed as a version of the bandwidth theorem of Bottcher, Schacht and Taraz in the setting of approximate decompositions. More precisely, let delta k be the infimum over all delta > 1/2 ensuring an approximate Kk-decomposition of any sufficiently large regular n-vertex graph G of degree at least delta n. Now suppose that G is an n-vertex graph which is close to r-regular for some r >(delta k+o(1))n and suppose that H1,MIDLINE HORIZONTAL ELLIPSIS,Ht is a sequence of bounded degree n-vertex k-chromatic separable graphs with n-ary sumation ie(Hi)<=(1-o(1))e(G). We show that there is an edge-disjoint packing of H1,MIDLINE HORIZONTAL ELLIPSIS,Ht into G. If the Hi are bipartite, then r >(1/2+o(1))n is sufficient. In particular, this yields an approximate version of the tree packing conjecture in the setting of regular host graphs G of high degree. Similarly, our result implies approximate versions of the Oberwolfach problem, the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree.