We experimentally and theoretically investigate the sources of phase noise for an injection-locked, diode-laser-pumped Nd:YAG laser. We use a fully quantum-mechanical model of the laser to describe the output phase noise of the laser explicitly in terms of the input noise sources. We compare the free-running and injection-locked output noise with the quantum-noise limit (QNL), and we find excellent quantitative agreement between the results of our experiments and theory. We show that the phase noise of the injection-locked laser can never be at the QNL for frequencies less than the injection-locking range. However, at frequencies well outside the linewidth of the slave laser, the phase noise can be at the QNL. We show that, although the technical noise of the laser system can be substantially reduced by injection locking, the influence of cavity-length fluctuations on the phase noise of an injection-locked laser is finite and much greater than the QNL. These fluctuations are the major impediment to achieving near-ideal performance for the injection-locked phase noise.