Even the most sensitive cosmic microwave background anisotropy experiments have signal-to-noise ratios <~5, so that an accurate determination of the properties of the cosmological signal requires a careful assessment of the experimental noise. Most of the experiments combine simultaneous multichannel observations, in which the presence of correlated noise is likely. This case is common for ground-based experiments, in which an important fraction of the noise could be atmospheric in origin. Here, the way to compute and determine the effects produced by this correlated noise is discussed; in particular, the paper considers the Tenerife experiments (three radiometers at 10, 15, and 33 GHz with two independent receivers each), showing how this effect has been taken into account properly in the more recent analysis of these data. It will be demonstrated that for each of the three radiometers of these experiments, the atmospheric noise is equivalent to a Gaussian noise common to both channels with a coherence time smaller than the binning time, the net effect being an enhancement of the error bars in the stacked scan as compared with the estimation for the case of pure uncorrelated noise. As expected from the spectral index of the atmosphere, the effect is more important at higher frequencies. The formalism is generalized and applied to the general case of simultaneous multichannel observations.