We study preferences over lotteries that pay a fixed prize at an uncertain future date: what we call time lotteries. The standard model of time preferences, Expected Discounted Utility, implies that individuals must be risk seeking towards such lotteries (RSTL). As a motivation, we show in an incentivized experiment that most subjects actually violate this property. We then make two theoretical contributions. First, we show that risk aversion over time lotteries can be captured by a generalization of Expected Discounted Utility that is obtained by keeping the postulates of Discounted Utility and Expected Utility. Second, we introduce a new property termed Stochastic Impatience, a risky counterpart of standard Impatience, and show that not only the model above, but also substantial generalizations that allow for non-Expected Utility and non-exponential discounting, cannot jointly accommodate it and even a single instance of risk aversion over time lotteries (or, equivalently, any violation of RSTL), showing a fundamental tension between the two.