Adding porosity to battery electrodes is sometimes useful for accommodating volumetric expansion, electrolyte access to active materials, or mitigating poor high-rate performance for thicker electrodes. Ordered macroporous electrode such as inverse opals, are a good model system: binder and conductive additive-free, interconnected electrically, have defined porosity consistent with thickness, good electrolyte wettability and surprisingly good behavior in half-cells and some Li-battery cells at normal rates. We show that at high charge and discharge rates, charge storage in macroporous electrode materials can be completely supressed, and then entirely recovered at low rates. Using a model system of inverse opal V2O5 in a flooded Li-battery three-electrode cell electrodes store almost no charge at rates >10 C, but capacity completely recovers when the rate is reduced to <1 C. We show how the IO material is modified under lithiation using X-ray diffraction, Raman scattering and electron microscopy. Chronoamperometric measurements together with a model to fit rate-dependent capacity decay suggests a dependence on the intrinsic out-of-plane conductivity of the electrode. The data show that electrodes with nanoscale dimensions and macroscale porosity are fundamentally limited for high-rate performance if the intrinsic electronic conductivity is poor, even when fully soaked with electrolyte.