Recent developments in large-scale machine learning suggest that by scaling up data, model size and training time properly, one might observe that improvements in pre-training would transfer favorably to most downstream tasks. In this work we systematically study this phenomena and establish that, as we increase the upstream accuracy, performance of downstream tasks \emph{saturates}. In particular, we investigate more than 4800 experiments on Vision Transformers, MLP-Mixers and ResNets with number of parameters ranging from ten million to ten billion, trained on the largest scale of available image data (JFT, ImageNet21K) and evaluated on more than 20 downstream image recognition tasks. We propose a model for downstream performance that reflects the saturation phenomena and captures the nonlinear relationship in performance of upstream and downstream tasks. Delving deeper to understand the reasons that give rise to these phenomena, we show that the observed saturation behavior is closely related to the way that representations evolve through the layers of the models. We showcase an even more extreme scenario where performance on upstream and downstream are at odds with each other. That is, in order to have a better downstream performance, we need to hurt upstream accuracy.