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In medical settings, Individual Variation (IV) refers to variation that is due not to population differences or errors, but rather to within-subject variation, that is the intrinsic and characteristic patterns of variation pertaining to a given instance or the measurement process. While taking into account IV has been deemed critical for proper analysis of medical data, this source of uncertainty and its impact on robustness have so far been neglected in Machine Learning (ML). To fill this gap, we look at how IV affects ML performance and generalization and how its impact can be mitigated. Specifically, we provide a methodological contribution to formalize the problem of IV in the statistical learning framework and, through an experiment based on one of the largest real-world laboratory medicine datasets for the problem of COVID-19 diagnosis, we show that: 1) common state-of-the-art ML models are severely impacted by the presence of IV in data; and 2) advanced learning strategies, based on data augmentation and data imprecisiation, and proper study designs can be effective at improving robustness to IV. Our findings demonstrate the critical relevance of correctly accounting for IV to enable safe deployment of ML in clinical settings.