This work demonstrates the first application of direct broadband (1 GHz-30 GHz) quality (Q) factor measurements of the uniform precession mode in magnetised garnet spheres for the accurate determination of the room-temperature intrinsic ferromagnetic linewidth (ΔH). The spheres were enclosed in a subwavelength cavity, so that the measured Q-factor depended mainly on their magnetic losses and the conduction losses of the cavity walls. The contribution of the latter is assessed by means of the recently proposed magnetic plasmon resonance model and has been found to be negligible. A total of 10 samples made from commercially available pure yttrium iron garnet (YIG) and gallium-substituted YIG have been measured, differing in diameter and/or saturation magnetisation Ms. The dependence of the intrinsic ΔH on the internal magnetic field is found to have near-perfect linear dependence, which cannot be said about the typically studied extrinsic ΔH even at high frequencies. It is found that the difference between the two linewidths, which becomes significant at low frequencies, can be attributed to a geometric effect. Due to its fundamental nature, this work is applicable not only to magnetic material characterization, but also to the study of the origins of losses in magnetic materials.