We report measurements of the atmospheric phase noise at 86 GHz, made by using the Hat Creek millimeter wavelength interferometer. The measurements used a dual-frequency technique to estimate the receiver noise contribution to the phase. The data, which cover projected baselines between 2000 and 40000 wavelengths, are used to derive median root mean squared (rms) phase noise as a function of baseline length. The implications of these results for the resolution limit to millimeter wavelength interferometry are discussed. We also discuss the time domain spectrum of atmospheric phase noise and consider an observing strategy that can mitigate the effects of adverse atmospheric conditions.