Camera-based physiological monitoring, especially remote photoplethysmography (rPPG), is a promising tool for health diagnostics, and state-of-the-art pulse estimators have shown impressive performance on benchmark datasets. We argue that evaluations of modern solutions may be incomplete, as we uncover failure cases for videos without a live person, or in the presence of severe noise. We demonstrate that spatiotemporal deep learning models trained only with live samples "hallucinate" a genuine-shaped pulse on anomalous and noisy videos, which may have negative consequences when rPPG models are used by medical personnel. To address this, we offer: (a) An anomaly detection model, built on top of the predicted waveforms. We compare models trained in open-set (unknown abnormal predictions) and closed-set (abnormal predictions known when training) settings; (b) An anomaly-aware training regime that penalizes the model for predicting periodic signals from anomalous videos. Extensive experimentation with eight research datasets (rPPG-specific: DDPM, CDDPM, PURE, UBFC, ARPM; deep fakes: DFDC; face presentation attack detection: HKBU-MARs; rPPG outlier: KITTI) show better accuracy of anomaly detection for deep learning models incorporating the proposed training (75.8%), compared to models trained regularly (73.7%) and to hand-crafted rPPG methods (52-62%).