A graph G = (V;E) is arbitrarily partitionable if for any sequence of posi- tive integers adding up to jV j, there is a sequence of vertex-disjoint subsets of V whose orders are given by , and which induce connected subgraphs. Such graph models, e.g., a computer network which may be arbitrarily par- titioned into connected subnetworks. In this paper we study the structure of such graphs and prove that unlike in some related problems, arbitrarily partitionable graphs may have arbitrarily many components after removing a cutset of a given size 2. The sizes of these components grow exponentially, though.