Large width limits have been a recent focus of deep learning research: modulo computational practicalities, do wider networks outperform narrower ones? Answering this question has been challenging, as conventional networks gain representational power with width, potentially masking any negative effects. Our analysis in this paper decouples capacity and width via the generalization of neural networks to Deep Gaussian Processes (Deep GP), a class of hierarchical models that subsume neural nets. In doing so, we aim to understand how width affects standard neural networks once they have sufficient capacity for a given modeling task. Our theoretical and empirical results on Deep GP suggest that large width is generally detrimental to hierarchical models. Surprisingly, we prove that even nonparametric Deep GP converge to Gaussian processes, effectively becoming shallower without any increase in representational power. The posterior, which corresponds to a mixture of data-adaptable basis functions, becomes less data-dependent with width. Our tail analysis demonstrates that width and depth have opposite effects: depth accentuates a model's non-Gaussianity, while width makes models increasingly Gaussian. We find there is a "sweet spot" that maximizes test set performance before the limiting GP behavior prevents adaptability, occurring at width = 1 or width = 2 for nonparametric Deep GP. These results make strong predictions about the same phenomenon in conventional neural networks: we show empirically that many neural network architectures need 10 - 500 hidden units for sufficient capacity - depending on the dataset - but further width degrades test performance.