Electron cyclotron emission correlation radiometry is used to measure the local turbulent electron temperature fluctuations in the plasma core of the Alcator C-Mod tokamak. Using standard analysis techniques, we see no evidence of the broadband fluctuations seen in other devices. From an analysis of the diagnostic resolution, coupled with these data, we can place an upper limit on the measurable fluctuation amplitude and set bounds for the wave number spectrum required for the electrostatic turbulence model of anomalous heat flux to be valid.