Hydrological drought, defined as below-average streamflow conditions, can be triggered by different mechanisms, which are to a large extent dictated by the climate. Moreover, the simulation of hydrological droughts highly depends on the model structure and how drought-triggering mechanisms are parameterized. In this large-sample hydrological study, we investigate how climate and model structure control hydrological drought simulations. We conducted sensitivity analysis on parameters of three frequently used hydrological models (HBV, SAC, and VIC) for the simulation of drought duration and drought deficit over 605 basins covering more than 10 different Köppen-Geiger climates. The sensitivity analysis revealed that, as anticipated, different parameters are sensitive in different climates. However, not all expected drought mechanisms were reflected in the parameter sensitivity: Especially, the sensitivity of ET parameters does not align with the theory, and the role of snow parameters in snow-related droughts shows a distinction between degree-day-based models and energy-balance models. Besides parameter sensitivity being different over climates, we also found that parameter sensitivity differed over the different models. Where HBV and SAC did display fairly similar behavior, in VIC other model mechanisms were triggered. This implies that conclusions on driving mechanisms in hydrological drought cannot be based on hydrological models only, as different models would lead to different conclusions. Hydrological models can have heuristic value in drought research, to formulate new theories and identify research directions, but formulated theories on driving processes should always be backed up by observations.