A recent X-ray observation of the cluster 1E 0657-56 (z = 0.296) with the Advanced Satellite for Cosmology and Astrophysics (ASCA) implied an unusually high temperature of ~17 keV. Such a high temperature would make it the hottest known cluster and would severely constrain cosmological models since, in a universe with critical density (Ω = 1), the probability of observing such a cluster is only ~4 × 10-5. Here we test the robustness of this observational result because it has such important implications. We analyzed the data using a variety of different data analysis methods and spectral analysis assumptions, and we found a temperature of ~11-12 keV in all cases, except for one class of spectral fits. These are fits in which the absorbing column density is fixed at the Galactic value. Using simulated data for a 12 keV cluster, we show that a high temperature of ~17 keV is artificially obtained if the true spectrum has a stronger low-energy cutoff than that for Galactic absorption only. The apparent extra absorption may be astrophysical in origin (either intrinsic or line of sight), or it may be a problem with the low-energy CCD efficiency. Although significantly lower than previous measurements, this temperature of kT ~ 11-12 keV is still relatively high since only a few clusters have been found to have temperatures higher than 10 keV, and the data therefore still present some difficulty for an Ω = 1 universe. Our results will also be useful to anyone who wants to estimate the systematic errors involved in the different methods of background subtraction of ASCA data for sources with a similar signal-to-noise ratio to that of the 1E 0657-56 data reported here.