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Recent advances in vision-and-language modeling have seen the development of Transformer architectures that achieve remarkable performance on multimodal reasoning tasks. Yet, the exact capabilities of these black-box models are still poorly understood. While much of previous work has focused on studying their ability to learn meaning at the word-level, their ability to track syntactic dependencies between words has received less attention. We take a first step in closing this gap by creating a new multimodal task targeted at evaluating understanding of predicate-noun dependencies in a controlled setup. We evaluate a range of state-of-the-art models and find that their performance on the task varies considerably, with some models performing relatively well and others at chance level. In an effort to explain this variability, our analyses indicate that the quality (and not only sheer quantity) of pretraining data is essential. Additionally, the best performing models leverage fine-grained multimodal pretraining objectives in addition to the standard image-text matching objectives. This study highlights that targeted and controlled evaluations are a crucial step for a precise and rigorous test of the multimodal knowledge of vision-and-language models.