This paper investigates the limitations of neural operators in learning solutions for the Hughes model, a first-order hyperbolic conservation law system for crowd dynamics. The model couples a Fokker–Planck equation, representing pedestrian density, with a Hamilton–Jacobi type equation (the eikonal equation). This Hughes model belongs to the class of nonlinear hyperbolic systems which often exhibit complex solution structures, including shocks and discontinuities. In this study, we assess the performance of three state of the art neural operators Fourier Neural Operator (FNO), Wavelet Neural Operator (WNO), and Multiwavelet Neural Operator (MWNO) in various challenging scenarios. Specifically, we consider (1) discontinuous and Gaussian initial conditions and (2) diverse boundary conditions while also examining the impact of different numerical schemes. Our results show that these neural operators perform well in easy scenarios characterized by fewer discontinuities in the initial condition, but, they struggle in complex scenarios with multiple initial discontinuities and dynamic boundary condition, even when trained specifically on such complex samples. The predicted solutions often appear smoother, resulting in a reduction in total variation and a loss of important physical features. This smoothing behavior is similar to issues discussed in Daganzo (1995), where models that introduce artificial diffusion were shown to miss essential features such as shock waves in hyperbolic systems. This suggests that current neural operator architectures may introduce unintended regularization effects that limit their ability to capture transport dynamics governed by discontinuities. Since the Hughes model shares important structural features with models used in traffic flow, these results also raise concerns about the ability of neural operator architectures to generalize to traffic applications where shock preservation is essential.