Keeping track of how states and relations of entities change as a text or dialog unfolds is a key prerequisite to discourse understanding. Despite this fact, there have been few systematic investigations into the ability of large language models (LLMs) to track discourse entities. In this work, we present a task to probe to what extent a language model can infer the final state of an entity given an English description of the initial state and a series of state-changing operations. We use this task to first investigate whether Flan-T5, GPT-3 and GPT-3.5 can track the state of entities, and find that only GPT-3.5 models, which have been pretrained on large amounts of code, exhibit this ability. We then investigate whether smaller models pretrained primarily on text can learn to track entities, through finetuning T5 on several training/evaluation splits. While performance degrades for more complex splits, we find that even for splits with almost no lexical overlap between training and evaluation, a finetuned model can often perform non-trivial entity tracking. Taken together, these results suggest that language models can learn to track entities but pretraining on large text corpora alone does not make this capacity surface.