In this paper we pursue the origin of the non-Gaussianity determined by a bispectrum analysis of the COBE Differential Microwave Radiometer (DMR) 4 yr sky maps. The robustness of the statistic is demonstrated by the rebinning of the data into 12 coordinate systems. By computing the bispectrum statistic as a function of various data partitions (by channel, frequency, and time interval), we show that the observed non-Gaussian signal is driven by the 53 GHz data. This frequency dependence strongly rejects the hypothesis that the signal is cosmological in origin. A jackknife analysis of the coadded 53 and 90 GHz sky maps reveals those sky pixels to which the bispectrum statistic is particularly sensitive. We find that by removing data from the 53 GHz sky maps for periods of time during which a known systematic effect perturbs the 31 GHz channels, the amplitudes of the bispectrum coefficients become completely consistent with that expected for a Gaussian sky. We conclude that the non-Gaussian signal detected by the normalized bispectrum statistic in the publicly available DMR sky maps is due to a systematic artifact. The impact of removing the affected data on estimates of the normalization of simple models of cosmological anisotropy is negligible.