Consider the robust network design problem of finding a minimum cost network with enough capacity to route all traffic demand matrices in a given polytope. We investigate the impact of different routing models in this robust setting: in particular, we compare oblivious routing, where the routing between each terminal pair must be fixed in advance, to dynamic routing, where routings may depend arbitrarily on the current demand. Our main result is a construction that shows that the optimal cost of such a network based on oblivious routing (fractional or integral) may be a factor of Ω(log n) more than the cost required when using dynamic routing. This is true even in the important special case of the asymmetric hose model. This answers a question in (Chekuri, SIGACT News 38(3):106-128, 2007), and is tight up to constant factors. Our proof technique builds on a connection between expander graphs and robust design for single-sink traffic patterns (Chekuri et al., Networks 50(1):50-54, 2007). © Springer Science+Business Media, LLC 2010.