Chain-of-thought (CoT) supervision has become a default assumption in fine-tuning large language models for reasoning tasks, with the intuition being that exposing models to intermediate reasoning steps leads to deeper learning than training on answers alone. We challenge this assumption. We introduce answer backtrackability, a measurable quantity capturing the degree to which a question can be reconstructed from its answer, and argue it is a key determinant of whether CoT supervision helps or hurts during fine-tuning. To test this, we construct two novel benchmarks derived from automata theory (NFA and DFA), where answers are transition tables from which the question is largely recoverable, and fine-tune three LLMs using LoRA on CoT and No-CoT versions of five datasets: GSM8K, MathInstruct, MATH, and our two formal-language benchmarks. On mathematical datasets, CoT consistently outperforms No-CoT. Within MathInstruct, Program-of-Thought (PoT) problems, where executable code answers are high-backtrackability, favour No-CoT, while CoT-format problems favour CoT. On our benchmarks, the pattern reverses: No-CoT outperforms CoT across all models and difficulty tiers. When the answer encodes sufficient structure to reconstruct the reasoning process, intermediate supervision adds noise rather than signal.