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One reason pretraining on self-supervised linguistic tasks is effective is that it teaches models features that are helpful for language understanding. However, we want pretrained models to learn not only to represent linguistic features, but also to use those features preferentially during fine-turning. With this goal in mind, we introduce a new English-language diagnostic set called MSGS (the Mixed Signals Generalization Set), which consists of 20 ambiguous binary classification tasks that we use to test whether a pretrained model prefers linguistic or surface generalizations during fine-tuning. We pretrain RoBERTa models from scratch on quantities of data ranging from 1M to 1B words and compare their performance on MSGS to the publicly available RoBERTa-base. We find that models can learn to represent linguistic features with little pretraining data, but require far more data to learn to prefer linguistic generalizations over surface ones. Eventually, with about 30B words of pretraining data, RoBERTa-base does demonstrate a linguistic bias with some regularity. We conclude that while self-supervised pretraining is an effective way to learn helpful inductive biases, there is likely room to improve the rate at which models learn which features matter.