Supervised fine-tuning with synthetic rationale data is widely assumed to improve language model performance on clinical prediction tasks by teaching models not just what to predict but why. We test this assumption on five-year Alzheimer's disease and related dementias (ADRD) prediction from longitudinal health histories. Across a large-scale controlled experiment of 504 configurations, we find that rationale-based SFT consistently and substantially hurts prediction performance relative to label-only fine-tuning. The degradation persists across model families and data scales, and is not resolved by using a reasoning-oriented base model. Crucially, the failure is not explained by poor rationale quality: human expert annotation confirms that the generated rationales are medically accurate and faithfully grounded in patient-specific evidence, and few-shot experiments show that the same rationales improve performance when used as inference-time demonstrations rather than training targets. We identify the root cause as a structural conflict between narrative plausibility and discriminative optimization. We hope our work paves the path toward a more precise understanding of when and how rationale-based supervision helps and when it does not, guiding the responsible development of language models for high-stakes clinical prediction.