  Many analyses of microwave background experiments neglect the correlation of
noise in different frequency or polarization channels. We show that these
correlations, should they be present, can lead to severe misinterpretation of
an experiment. In particular, correlated noise arising from either electronics
or atmosphere may mimic a cosmic signal. We quantify how the likelihood
function for a given experiment varies with noise correlation, using both
simple analytic models and actual data. For a typical microwave background
anisotropy experiment, noise correlations at the level of 1\% of the overall
noise can seriously {\it reduce} the significance of a given detection.
