Many analyses of microwave background experiments neglect the correlat
ion of noise in different frequency or polarization channels. We show
that these correlations, should they be present, can lead to severe mi
sinterpretation of an experiment. In particular, correlated noise aris
ing from either electronics or atmosphere may mimic a cosmic signal. W
e quantify how the likelihood function for a given experiment varies w
ith noise correlation, using both simple analytic models and actual da
ta. For a typical microwave background anisotropy experiment, noise co
rrelations at the level of 1% of the overall noise can seriously reduc
e the significance of a given detection.