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The Boolean lattice (2[n],subset of) is the family of all subsets of [n]={1,MIDLINE HORIZONTAL ELLIPSIS,n}, ordered by inclusion. Let P be a partially ordered set. We prove that if n is sufficiently large, then there exists a packing P of copies of P in (2[n],subset of) that covers almost every element of 2[n]: 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 2n-2, then the truncated Boolean lattice 2[n]-{ null ,[n]} can be partitioned into copies of P. This confirms a conjecture of Lonc from 1991.