In this paper, we study a static traffic assignment that accounts for the boundedly rational route choice behavior of travelers. This assignment induces uncertainties to the ex-ante evaluation of a policy measure: the boundedly rational assignment is non-unique and the indifference band is an uncertain parameter. We consider two different ways to model the optimization problem that finds the best and worst-performing Boundedly Rational User Equilibrium with respect to the total travel time (Best/Worst-case BRUE). The first is the so-called branch approach, the second is a bilevel model. The latter approach is better suited to exploit techniques from parametric optimization and enables us, e.g., to prove the continuity of the optimal value function corresponding to the Best/Worst-case BRUE with respect to perturbations in the indifference band. We report on some numerical experiments. In addition, we extend our results to the Network Design Problem: we prove the existence of a second-best toll pricing scheme under bounded rationality.