In order to extract full cosmological information from next-generation large and high-precision weak lensing (WL) surveys (e.g., Euclid, Roman, and LSST), higher-order statistics that probe the small-scale, nonlinear regime of large-scale structure (LSS) need to be utilized. WL peak counts, which trace overdensities in the cosmic web, are one promising and simple statistic for constraining cosmological parameters. The physical origin of WL peaks have previously been linked to dark matter halos along the line of sight, and this peak-halo connection has been used to develop various semianalytic halo-based models for predicting peak counts. Here, we study the origin of WL peaks and the effectiveness of halo-based models for WL peak counts using a suite of ray-tracing N-body simulations. We compare WL peaks in convergence maps from the full simulations to those in maps created from only particles associated with halos—the latter playing the role of a "perfect" halo model. We find that, while halo-only contributions are able to replicate peak counts qualitatively well, halos do not explain all WL peaks. Halos particularly underpredict negative peaks, which are associated with local overdensities in large-scale underdense regions along the line of sight. In addition, neglecting nonhalo contributions to peaks counts leads to a significant bias on the parameters (Ωm, σ8) for surveys larger than ⪆100 deg2 . We conclude that other elements of the cosmic web, outside and far away from dark matter halos, need to be incorporated into models of WL peaks in order to infer unbiased cosmological constraints.