Fay, Hurlbert and Tennant recently introduced a one-player game on a finite connected graph G, which they called cup stacking. Stacks of cups are placed at the vertices of G, and are transferred between vertices via stacking moves, subject to certain constraints, with the goal of stacking all cups at a single target vertex. If this is possible for every target vertex of G, then G is called stackable. In this paper, we prove that if G admits a Hamilton path, then G is stackable, which confirms several of the conjectures raised by Fay, Hurlbert and Tennant. Furthermore, we prove stackability for certain powers of bipartite graphs, and we construct graphs of arbitrarily large minimum degree and connectivity that do not allow stacking onto any of their vertices.