We present measurements of the resolving power of a soft x-ray spectrometer consisting of 200 nm period lightweight, alignment-insensitive critical-angle transmission (CAT) gratings and a lightweight slumped-glass Wolter-I focusing mirror pair. We measure and model contributions from source, mirrors, detector pixel size, and grating period variation to the natural linewidth spectrum of the Al-K α 1 α 2 doublet. Measuring up to the 18th diffraction order, we consistently obtain small broadening due to gratings corresponding to a minimum effective grating resolving power Rg>10,000 with 90% confidence. Upper limits are often compatible with Rg=∞. Independent fitting of different diffraction orders, as well as ensemble fitting of multiple orders at multiple wavelengths, gives compatible results. Our data leads to uncertainties for the Al-Kα doublet linewidth and line separation parameters two to three times smaller than values found in the literature. Data from three different gratings are mutually compatible. This demonstrates that CAT gratings perform in excess of the requirements for the Arcus Explorer mission and are suitable for next-generation space-based x-ray spectrometer designs with resolving power five to 10 times higher than the transmission grating spectrometer onboard the Chandra X-ray Observatory.