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It has been suggested by Hogan (1992) that the microwave background anisotropy detected by the COBE DMR experiment (Smoot, et al 1992) might be produced by inverse Compton scattering from hot diffuse clouds of electrons in nearby superclusters. If the COBE fluctuations are due to this mechanism then the absence of anticorrelations between maps of the cosmic microwave and cosmic X-ray backgrounds constrains the temperature (kT $\geq$ 30 keV) and density (n $\leq$ 5 x 10$^{-5}$ cm$^{-3}$) of the ionized supercluster gas. Since the lack of distortion of the spectrum of the microwave background constrains the temperature of the IGM to be less than 10 keV (Wright, et al 1993), we conclude that the fluctuations observed by COBE are not due to this mechanism. This work was supported in part by NSF grant AST-8914988 and NASA grant NAF 5-1713.