Many previous studies have used emission line or broadband filter ratios to infer the presence of temperature gradients in the quiet solar corona. Recently it has been suggested that these temperature gradients are not real, but result from the superposition of isothermal loops with different temperatures and density scale heights along the line of sight. A model describing this hydrostatic weighting bias has been developed by Aschwanden & Acton. In this paper we present the application of the Aschwanden & Acton differential emission measure model to Solar and Heliospheric Observatory Solar Ultraviolet Measurement of Emitted Radiation (SUMER) observations of a quiet coronal streamer. Simultaneous Yohkoh soft X-ray telescope (SXT) observations show increases in the filter ratios with height above the limb, indicating an increase in temperature. The application of the Aschwanden & Acton model to these SUMER data, however, show that the temperature is constant with height and that the distribution of temperatures in the corona is much too narrow for the hydrostatic weighting bias to have any effect on the SXT filter ratios. We consider the possibility that there is a tenuous hot component (~3 MK) that accounts for the SXT observations. We find that a hot plasma with an emission measure sufficient to reproduce the observed SXT fluxes would also produce significant count rates in the high-temperature emission lines in the SUMER wavelength range. These lines are not observed, and we conclude that the SUMER spectra are not consistent with the SXT filter ratio temperatures. Calculations from a hydrodynamic loop model suggest that nonuniform footpoint heating may be consistent with the temperatures and densities observed at most heights, consistent with the recent analysis of relatively cool (~1 MK) active region loops. We also find, however, that at the lowest heights the observed densities are smaller than those predicted by uniform or footpoint heating.