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Deep graph neural networks (GNNs) have become a cornerstone of computational materials engineering because they can learn representations directly from crystal graphs. Yet a fundamental failure mode remains hidden when these models are applied to thermal transport: over-smoothing. After only a modest number of message-passing layers, node features converge to a single global vector, erasing the atom-resolved local environments that govern phonon scattering. The problem is dramatically amplified in large unit cells—structures containing 50 or more atoms, common in skutterudites, clathrates, and complex perovskites—because the graph diameter forces the model designer to choose between insufficient propagation depth (missing long-range interactions) and excessive depth (catastrophic smoothing). Lattice thermal conductivity κ emerges from local force constants, anharmonicity, and defect-sensitive scattering; over-smoothing replaces these variations with a featureless average, producing systematically erroneous predictions. This failure-mode analysis demonstrates that the central conflict is structural: thermal transport demands precisely the short- to medium-range distinctions that deep GNNs destroy. Four distinct failure modes are identified and mechanistically linked to the over-smoothing process. Detection principles based on feature similarity, depth sensitivity, and defect response are proposed. Mitigation strategies grounded in architectural modifications—residual connections, normalization, multi-scale pathways, and attention—are shown to preserve local information without sacrificing global coherence. The analysis reveals why standard crystal GNN architectures succeed for formation energies or band gaps yet fail systematically for thermal transport in realistic materials. By reframing over-smoothing as the dominant limitation rather than a minor training artifact, this work provides a conceptual roadmap for next-generation GNNs tailored to phonon physics.