Multimodal deep learning systems are deployed in dynamic scenarios due to the robustness afforded by multiple sensing modalities. Nevertheless, they struggle with varying compute resource availability (due to multi-tenancy, device heterogeneity, etc.) and fluctuating quality of inputs (from sensor feed corruption, environmental noise, etc.). Statically provisioned multimodal systems cannot adapt when compute resources change over time, while existing dynamic networks struggle with strict compute budgets. Additionally, both systems often neglect the impact of variations in modality quality. Consequently, modalities suffering substantial corruption may needlessly consume resources better allocated towards other modalities. We propose ADMN, a layer-wise Adaptive Depth Multimodal Network capable of tackling both challenges: it adjusts the total number of active layers across all modalities to meet strict compute resource constraints and continually reallocates layers across input modalities according to their modality quality. Our evaluations showcase ADMN can match the accuracy of state-of-the-art networks while reducing up to 75% of their floating-point operations.