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Rotational invariance, a core design principle in graph neural networks (GNNs) for materials property prediction, has enabled remarkable accuracy for scalar quantities such as formation energy and band gap yet becomes a fundamental liability when the target property is a tensor of rank two or higher. Anisotropic elastic constants, which describe the directional stiffness of crystals and enter engineering design as a 6×6 Voigt matrix with up to 21 independent components in triclinic systems, must transform covariantly under rotation; any architecture that collapses all orientational information into a single invariant scalar cannot, in principle, recover the required tensorial structure. This failure-mode analysis articulates four distinct mechanisms by which invariance induces systematic error—diagonal indistinguishability, off-diagonal blindness, shear underspecification, and symmetry-induced degeneracy—and demonstrates how each arises directly from the interplay of global pooling, distance-only edge features, and symmetry averaging that characterize rotationally invariant models such as CGCNN, MEGNet, and SchNet. Detection principles based on equality tests, anisotropy-ratio checks, and differential error audits provide practitioners with practical diagnostics, while mitigation principles centered on equivariant architectures, directional encoding, and stress-tensor multi-task learning chart a clear path toward reliable tensorial prediction. The implications extend beyond elasticity to any directional property in low-symmetry crystals and call for a principled redesign of materials GNNs that treats invariance as a deliberate choice rather than a default.