Monitoring vertical land motions (VLMs) at the level of 0.1 mm/yr remains one of the most challenging scientific applications of global navigation satellite systems (GNSS). Such small rates of change can result from climatic and tectonic phenomena, and their detection is important to many solid Earth-related studies, including the prediction of coastal sea-level change and the understanding of intraplate deformation. Reaching a level of precision allowing to detect such small signals requires a thorough understanding of the stochastic variability in GNSS VLM time series. This paper investigates how the aperiodic part of non-tidal atmospheric and oceanic loading (NTAOL) deformations influences the stochastic properties of VLM time series. Using the time series of over 10,000 stations, we describe the impact of correcting for NTAOL deformation on 5 complementary metrics, namely: the repeatability of position residuals, the power-spectrum of position residuals, the estimated time-correlation properties, the corresponding velocity uncertainties, and the spatial correlation of the residuals. We show that NTAOL deformations cause a latitude-dependent bias in white noise plus power-law model parameter estimates. This bias is significantly mitigated when correcting for NTAOL deformation, which reduces velocity uncertainties at high latitudes by 70%. Therefore, removing NTAOL deformation before the statistical analysis of VLM time series might help to detect subtle VLM signals in these areas. Our spatial correlation analysis also reveals a seasonality in the spatial correlation of the residuals, which is reduced after removing NTAOL deformation, confirming that NTAOL is a clear source of common-mode errors in GNSS VLM time series.