Numerous studies have shown that multimodal LLMs process speech and images well but fail in non-intuitive ways rendering trivial tasks such as object counting unreliable. We investigate this behavior from an information-theoretic perspective by framing multimodal LLM inference as a mismatched decoder problem: a decoder trained primarily on text can only extract information along text-aligned directions (removing up to 98% of the variation in modality-specific (non-text) directions improves decoder loss) and the amount of accessible information is bounded by the Generalized Mutual Information (GMI). We show that information loss is bounded as the distributional mismatch between the source data and the text data increases, and as the sensitivity of the decoder increases. This bound is a function of the model's scoring rule not its architecture. We validate the predictions across five models spanning speech and vision. A controlled study (two Prismatic VLMs differing only in encoder text-alignment) shows that the bottleneck lies in the scoring rule of the decoder rather than the text-alignment of the encoder or the learned projection. A LoRA intervention demonstrates that simply training with an emotion-related objective improves emotion detection from 17.3% to 61.8% task accuracy without affecting other attributes, confirming that the training objective determines what becomes accessible.