We consider a discrete-time nonatomic routing game with variable demand and uncertain costs. Given a routing network with single origin and destination, the costs functions on edges depend on some uncertain persistent state parameter. Every period, a variable traffic demand routes through the network. The experienced costs are publicly observed and the belief about the state parameter is Bayesianly updated. This paper studies the dynamics of equilibrium and beliefs. We say that there is strong learning when beliefs converge to the truth and there is weak learning when equilibrium flows converge to those under complete information. Our main result is a characterization of the networks for which learning occurs for all increasing cost functions, given highly variable demand. We prove that these networks have a series-parallel structure and provide a counterexample to prove that the condition is necessary.