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Precision optical measurements of the electron-spin precession of nitrogen-vacancy (NV) centers in diamond form the basis of numerous applications. The most sensitivity-demanding applications, such as femtotesla magnetometry, require the ability to measure changes in GHz spin transition frequencies at the sub-millihertz level, corresponding to a fractional resolution of better than ${10}^{-12}$. Here we study the impact of microwave (MW) phase noise on the response of an NV sensor. Fluctuations of the phase of the MW waveform cause undesired rotations of the NV spin state. These fluctuations are imprinted in the optical readout signal and, left unmitigated, are indistinguishable from magnetic-field noise. We show that the phase noise of several common commercial MW generators results in an effective $\mathrm{pT}{\mathrm{s}}^{1/2}$-range noise floor that varies with the MW carrier frequency and the detection frequency of the pulse sequence. The data are described by a frequency-domain model incorporating the MW phase-noise spectrum and the filter-function response of the sensing protocol. For controlled injection of white and random-walk phase noise, the observed NV magnetic noise floor is described by simple analytic expressions that accurately capture the scaling with pulse sequence length and the number of $\pi$pulses. We outline several strategies to suppress the impact of MW phase noise and implement a version, based on gradiometry, that realizes a $>10$-fold suppression. Our study highlights an important challenge in the pursuit of sensitive diamond quantum sensors and is applicable to other qubit systems with a large transition frequency. Published by the American Physical Society2024