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Inspired from human cognition, machine learning systems are gradually revealing advantages of sparser and more modular architectures. Recent work demonstrates that not only do some modular architectures generalize well, but they also lead to better out-of-distribution generalization, scaling properties, learning speed, and interpretability. A key intuition behind the success of such systems is that the data generating system for most real-world settings is considered to consist of sparsely interacting parts, and endowing models with similar inductive biases will be helpful. However, the field has been lacking in a rigorous quantitative assessment of such systems because these real-world data distributions are complex and unknown. In this work, we provide a thorough assessment of common modular architectures, through the lens of simple and known modular data distributions. We highlight the benefits of modularity and sparsity and reveal insights on the challenges faced while optimizing modular systems. In doing so, we propose evaluation metrics that highlight the benefits of modularity, the regimes in which these benefits are substantial, as well as the sub-optimality of current end-to-end learned modular systems as opposed to their claimed potential.