  Let $P$ be a partially ordered set. We prove that if $n$ is sufficiently
large, then there exists a packing $\mathcal{P}$ of copies of $P$ in the
Boolean lattice $(2^{[n]},\subset)$ that covers almost every element of
$2^{[n]}$: $\mathcal{P}$ might not cover the minimum and maximum of $2^{[n]}$,
and at most $|P|-1$ additional points due to divisibility. In particular, if
$|P|$ divides $2^{n}-2$, then the truncated Boolean lattice
$2^{[n]}-\{\emptyset,[n]\}$ can be partitioned into copies of $P$. This
confirms a conjecture of Lonc from 1991.
