  Let $\mathcal G$ be a separable family of graphs. Then for all positive
constants $\epsilon$ and $\Delta$ and for every sufficiently large integer $n$,
every sequence $G_1,\dotsc,G_t\in\mathcal G$ of graphs of order $n$ and maximum
degree at most $\Delta$ such that $e(G_1)+\dotsb+e(G_t) \leq
(1-\epsilon)\binom{n}{2}$ packs into $K_n$. This improves results of
B\"ottcher, Hladk\'y, Piguet, and Taraz when $\mathcal G$ is the class of trees
and of Messuti, R\"odl, and Schacht in the case of a general separable family.
The result also implies approximate versions of the Oberwolfach problem and of
the Tree Packing Conjecture of Gy\'arf\'as (1976) for the case that all trees
have maximum degree at most $\Delta$. The proof uses the local resilience of
random graphs and a special multi-stage packing procedure.
