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The deployment of machine learning models in safety-critical applications comes with the expectation that such models will perform well over a range of contexts (e.g., a vision model for classifying street signs should work in rural, city, and highway settings under varying lighting/weather conditions). However, these one-size-fits-all models are typically optimized for average case performance, encouraging them to achieve high performance in nominal conditions but exposing them to unexpected behavior in challenging or rare contexts. To address this concern, we develop a new method for training context-dependent models. We extend Bridge-Mode Connectivity (BMC) (Garipov et al., 2018) to train an infinite ensemble of models over a continuous measure of context such that we can sample model parameters specifically tuned to the corresponding evaluation context. We explore the definition of context in image classification tasks through multiple lenses including changes in the risk profile, long-tail image statistics/appearance, and context-dependent distribution shift. We develop novel extensions of the BMC optimization for each of these cases and our experiments demonstrate that model performance can be successfully tuned to context in each scenario.