The magnitude of spin accumulation created in semiconductors by electrical injection of spin-polarized electrons from a ferromagnetic tunnel contact is investigated, focusing on how the spin signal detected in a Hanle measurement varies with the thickness of the tunnel oxide. An extensive set of spin-transport data for Si and Ge magnetic tunnel devices reveals a scaling with the tunnel resistance that violates the core feature of available theories, namely, the linear proportionality of the spin voltage to the injected spin current density. Instead, an anomalous scaling of the spin signal with the tunnel resistance is observed, following a power law with an exponent between 0.75 and 1 over 6 decades. The scaling extends to tunnel resistance values larger than 10^9 Ω μm2, far beyond the regime where the classical impedance mismatch or back flow into the ferromagnet play a role. This scaling is incompatible with existing theory for direct tunnel injection of spins into the semiconductor. It also demonstrates conclusively that the large spin signal does not originate from two-step tunneling via localized states near the oxide/semiconductor interface. Control experiments show that spin accumulation in localized states within the tunnel barrier or artifacts are also not responsible. Altogether, the scaling results suggest that, contrary to all existing descriptions, the spin signal is proportional to the applied bias voltage, rather than the (spin) current.