Let @math be a partially ordered set. We prove that if @math is sufficiently large, then there exists a packing @math of copies of @math in the Boolean lattice @math that covers almost every element of @math : @math might not cover the minimum and maximum of @math , and at most @math additional points due to divisibility. In particular, if @math divides @math , then the truncated Boolean lattice @math can be partitioned into copies of @math . This confirms a conjecture of Lonc from 1991.
 Let @math be a partially ordered set. If the Boolean lattice @math can be partitioned into copies of @math for some positive integer @math , then @math must satisfy the following two trivial conditions: (1) the size of @math is a power of @math , (2) @math has a unique maximal and minimal element. Resolving a conjecture of Lonc, it was shown by Gruslys, Leader and Tomon that these conditions are sufficient as well. In this paper, we show that if @math only satisfies condition (2), we can still almost partition @math into copies of @math . We prove that if @math has a unique maximal and minimal element, then there exists a constant @math such that all but at most @math elements of @math can be covered by disjoint copies of @math . Let @math be a poset of size @math that has a greatest and a least element. We prove that, for sufficiently large @math , the Boolean lattice @math can be partitioned into copies of @math . This resolves a conjecture of Lonc. Let @math be an induced subgraph of the hypercube @math for some @math . We show that if @math is a power of @math then, for sufficiciently large @math , the vertex set of @math can be partitioned into induced copies of @math . This answers a question of Offner. In fact, we prove a stronger statement: if @math is a subset of @math for some @math and if @math is a power of @math , then, for sufficiently large @math , @math can be partitioned into isometric copies of @math . Let @math be an induced subgraph of the torus @math . We show that when @math is even and @math divides some power of @math , then for sufficiently large @math the torus @math has a perfect vertex-packing with induced copies of @math . On the other hand, disproving a conjecture of Gruslys, we show that when @math is odd and not a prime power, then there exists @math such that @math divides some power of @math , but there is no @math such that @math has a perfect vertex-packing with copies of @math . We also disprove a conjecture of Gruslys, Leader and Tan by exhibiting a subgraph @math of the @math -dimensional hypercube @math , such that there is no @math for which @math has a perfect edge-packing with copies of @math . Let @math be an induced subgraph of the hypercube @math , for some @math . We show that for some @math , the vertices of @math can be partitioned into induced copies of @math and a remainder of at most @math vertices. We also show that the error term cannot be replaced by anything smaller than @math .