Languages are governed by syntactic constraints—structural rules that determine which sentences are grammatical in the language. In English, one such constraint is subject-verb agreement, which dictates that the number of a verb must match the number of its corresponding subject: “the dogs run”, but “the dog runs”. While this constraint appears to be simple, in practice speakers make agreement errors, particularly when a noun phrase near the verb differs in number from the subject (for example, a speaker might produce the ungrammatical sentence “the key to the cabinets are rusty”). This phenomenon, referred to as agreement attraction, is sensitive to a wide range of properties of the sentence; no single existing model is able to generate predictions for the wide variety of materials studied in the human experimental literature. We explore the viability of neural network language models—broad-coverage systems trained to predict the next word in a corpus—as a framework for addressing this limitation. We analyze the agreement errors made by Long Short-Term Memory (LSTM) networks and compare them to those of humans. The models successfully simulate certain results, such as the so-called number asymmetry and the difference between attraction strength in grammatical and ungrammatical sentences, but failed to simulate others, such as the effect of syntactic distance or notional (conceptual) number. We further evaluate networks trained with explicit syntactic supervision, and find that this form of supervision does not always lead to more human-like syntactic behavior. Finally, we show that the corpus used to train a network significantly affects the pattern of agreement errors produced by the network, and discuss the strengths and limitations of neural networks as a tool for understanding human syntactic processing.