A possible explanation for the impressive performance of masked language model (MLM) pre-training is that such models have learned to represent the syntactic structures prevalent in classical NLP pipelines. In this paper, we propose a different explanation: MLMs succeed on downstream tasks almost entirely due to their ability to model higher-order word co-occurrence statistics. To demonstrate this, we pre-train MLMs on sentences with randomly shuffled word order, and show that these models still achieve high accuracy after fine-tuning on many downstream tasks---including tasks specifically designed to be challenging for models that ignore word order. Our models perform surprisingly well according to some parametric syntactic probes, indicating possible deficiencies in how we test representations for syntactic information. Overall, our results show that purely distributional information largely explains the success of pre-training, and underscore the importance of curating challenging evaluation datasets that require deeper linguistic knowledge.