Sequential robot manipulation requires policies to execute novel combinations of familiar instruction components. However, collecting demonstrations for all possible instruction tuples is combinatorially expensive, while sparsely covered datasets often fail under out-of-distribution recombination. This paper studies compositional generalization through the lens of instruction-space coverage. We decompose the generalization gap into three sources: \textit{marginal instruction shift}, \textit{instruction-compositional shift}, and \textit{context--action shift}. This decomposition allows us to diagnose when sparse training coverage is sufficient, and what structure the training set must preserve for reliable action prediction. Our results show that exhaustive tuple enumeration is unnecessary: a structured subset can recover strong out-of-distribution performance when it covers action-relevant dependencies. We further find that sparse training often fails due to instruction steering rather than missing low-level skills; finetuning with only one demonstration per task improves OOD success from \(0.4\%\) to \(54.7\%\). For semantically dependent tasks, effective coverage must capture relational structure rather than only factor diversity. These findings suggest that efficient robot data collection should prioritize dependency coverage in instruction space over exhaustive task expansion. Additional results, code, and dataset are available on the \href{https://yixiaowang7.github.io/Diagnosing_Compositional_Generalization_Robot_Page/}{project website}.

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