We say that two hypergraphs H1 and H2 with v vertices each can be packed if there are edge disjoint hypergraphs H1′and H2′on the same set V of v vertices, where Hi′is isomorphic to Hi.It is shown that for every fixed integers k and t, where t≤k≤2t-2 and for all sufficiently large v there are two (t, k, v) partial designs that cannot be packed. Moreover, there are two isomorphic partial (t, k, v)-designs that cannot be packed. It is also shown that for every fixed k≥2t-1 and for all sufficiently large v there is a (λ1, t,k,v) partial design and a (λ1, t,k,v) partial design that cannot be packed, where λ1 λ2≤O(vk-2t+1log v). Both results are nearly optimal asymptotically and answer questions of Teirlinck. The proofs are probabilistic.