Maintaining high resolving power is a primary challenge in hard x-ray spectroscopy of newly developed bright and transient x-ray sources such as laser-produced plasmas. To address this challenge, the line widths in x-ray spectra with energies in the 17 keV to 70 keV range were recorded by positioning the detectors on and behind the focal circles of Cauchois type transmission-crystal spectrometers. To analyze and understand the observed line widths, we developed a geometrical model that accounts for source broadening and various instrumental broadening mechanisms. The x-ray sources were laboratory Mo or W electron-bombarded anodes, and the spectra were recorded on photostimulable phosphor image plates. For these relatively small x-ray sources, it was found that when the detector was placed on or near the focal circle, the line widths were dominated by the effective spatial resolution of the detector. When the detector was positioned beyond the focal circle, the line widths were determined primarily by source-size broadening. Moreover, the separation between the spectral lines increased with distance behind the focal circle faster than the line widths, resulting in increased resolving power with distance. Contributions to line broadenings caused by the crystal thickness, crystal rocking curve width, geometrical aberrations, and natural widths of the x-ray transitions were in all cases smaller than detector and source broadening, but were significant for some spectrometer geometries. The various contributions to the line widths, calculated using simple analytical expressions, were in good agreement with the measured line widths for a variety of spectrometer and source conditions. These modeling and experimental results enable the design of hard x-ray spectrometers that are optimized for high resolving power and for the measurement of the x-ray source size from the line widths recorded behind the focal circle.