Tiling the Boolean lattice with copies of a poset

Let P be a partially ordered set with a unique maximal and minimal element, and size 2m, where m is a positive integer. Settling a conjecture of Lonc, we prove that if n is sufficiently large, then the Boolean lattice 2[n] can be partitioned into isomorphic copies of P. Also, we show that if P has a unique maximum and minimum, but the size of P not necessarily a power of 2, then there exists a constant c = c(P) such that all but at most c elements of 2[n] can be covered by disjoint copies of P.