Data visualizations are vital components of many scientific articles and news stories. Current vision-language models (VLMs) still struggle on basic data visualization understanding tasks, but the causes of failure remain unclear. Are VLM failures attributable to limitations in how visual information in the data visualization is encoded, how information is transferred between the vision and language modules, or how information is processed within the language module? We developed FUGU, a suite of data visualization understanding tasks, to precisely characterize potential sources of difficulty (e.g., extracting the position of data points, distances between them, and other summary statistics). We used FUGU to investigate three widely used VLMs. To diagnose the sources of errors produced by these models, we used activation patching and linear probes to trace information flow through models across a variety of prompting strategies. We found that some models fail to generate the coordinates of individual data points correctly, and these initial errors often lead to erroneous final responses. When these models are provided with the correct coordinates, performance improves substantially. Moreover, even when the model generates an incorrect response, the correct coordinates can be successfully read out from the latent representations in the vision encoder, suggesting that the source of these errors lies in the vision-language handoff. We further found that while providing correct coordinates helps with tasks involving one or a small number of data points, it generally worsens performance for tasks that require extracting statistical relationships across many data points. Fine-tuning models on FUGU also fails to yield ceiling performance. These findings point to architectural constraints in current VLMs that might pose significant challenges for reliable data visualization understanding.