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Recent papers have appeared describing certain peculiarities of measures of quantum-mechanical entanglement in local quantum field theories with nonvanishing gravitational anomaly. We show that these peculiarities follow from the fact that the Hilbert space of a gravitationally anomalous QFT simply does not ever tensor factorize at all into Hilbert spaces in complementary regions, and thus the defining assumption on which the concept of quantum entanglement is based, always fails in gravitationally anomalous theories. We show the breakdown of all known definitions of entanglement when the gravitationally anomaly is present, and give a general proof that no such definition exists, based on a series of tight logical connections between tensor factorizations, boundary conditions, and the gravitational anomaly