In this talk, we discuss two failure cases of common practices that are typically believed to improve on vanilla methods: (i) adversarial training can lead to worse robust accuracy than standard training (ii) active learning can lead to a worse classifier than a model trained using uniform samples. In particular, we can prove both mathematically and empirically, that such failures can happen in the small-sample regime. We discuss high-level explanations derived from the theory, that shed light on the causes of these phenomena in practice.