Motivated by the need for reliable traffic management under fluctuating travel demand, we study the problem of determining the worst-case congestion in a multi-commodity traffic network subject to demand uncertainty. To this end, we stress-test a given network by identifying demand realizations and corresponding travelers’ route choices that maximize congestion. The users of the traffic network are assumed to act according to one of the two Wardrop principles—the user equilibrium or the system optimum—so that the resulting congestion models can be seen as bilevel problems with a single leader and multiple followers. To address uncertain travel demand, we consider different models such as ellipsoidal or budgeted uncertainty sets and the hose polyhedron. We present single-level mixed-integer nonlinear reformulations of the congestion models that exploit binary variables and big-M constants, prove the existence of optimal solutions, derive valid big-Ms, and propose several enhancement techniques to further strengthen the formulations. An extensive computational study on instances of the Sioux Falls network and instances from the SNDlib demonstrates the computational effectiveness of the proposed techniques and provides insight into the impact of different congestion measures and uncertainty models on the resulting worst-case congestion.