Building on Pomatto, Strack, and Tamuz (2020), we identify a tight condition for when background risk can induce first-order stochastic dominance. Using this condition, we show that under plausible levels of background risk, no theory of choice under risk can simultaneously satisfy the following three economic postulates: (i) Decision makers are risk-averse over small gambles, (ii) their preferences respect stochastic dominance, and (iii) they account for background risk. This impossibility result applies to expected utility theory, prospect theory, rank-dependent utility, and many other models.