Connected and automated vehicles (CAVs) are expected to yield significant improvements in safety, energy efficiency, and time utilization. However, their net effect on energy and environmental outcomes is unclear. Higher fuel economy reduces the energy required per mile of travel, but it also reduces the fuel cost of travel, incentivizing more travel and causing an energy "rebound effect." Moreover, CAVs are predicted to vastly reduce the time cost of travel, inducing further increases in travel and energy use. In this paper, we forecast the induced travel and rebound from CAVs using data on existing travel behavior. We develop a microeconomic model of vehicle miles traveled (VMT) choice under income and time constraints; then we use it to estimate elasticities of VMT demand with respect to fuel and time costs, with fuel cost data from the 2017 United States National Household Travel Survey (NHTS) and wage-derived predictions of travel time cost. Our central estimate of the combined price elasticity of VMT demand is -0.4, which differs substantially from previous estimates. We also find evidence that wealthier households have more elastic demand, and that households at all income levels are more sensitive to time costs than to fuel costs. We use our estimated elasticities to simulate VMT and energy use impacts of full, private CAV adoption under a range of possible changes to the fuel and time costs of travel. We forecast a 2-47% increase in travel demand for an average household. Our results indicate that backfire - i.e., a net rise in energy use - is a possibility, especially in higher income groups. This presents a stiff challenge to policy goals for reductions in not only energy use but also traffic congestion and local and global air pollution, as CAV use increases.