As multimodal learning finds applications in a wide variety of high-stakes societal tasks, investigating their robustness becomes important. Existing work has focused on understanding the robustness of vision-and-language models to imperceptible variations on benchmark tasks. In this work, we investigate the robustness of multimodal classifiers to cross-modal dilutions – a plausible variation. We develop a model that, given a multimodal (image + text) input, generates additional dilution text that (a) maintains relevance and topical coherence with the image and existing text, and (b) when added to the original text, leads to misclassification of the multimodal input. Via experiments on Crisis Humanitarianism and Sentiment Detection tasks, we find that the performance of task-specific fusion-based multimodal classifiers drops by 23.3% and 22.5%, respectively, in the presence of dilutions generated by our model. Metric-based comparisons with several baselines and human evaluations indicate that our dilutions show higher relevance and topical coherence, while simultaneously being more effective at demonstrating the brittleness of the multimodal classifiers. Our work aims to highlight and encourage further research on the robustness of deep multimodal models to realistic variations, especially in human-facing societal applications.