We present a novel and general technique for automated test generation that combines tight bounds with incremental SAT solving. The proposed technique uses incremental SAT to build test suites targeting a specific testing criterion, amongst various black-box and white-box criteria. As our experimental results show, the combination of tight bounds with incremental SAT, and the testing criterion driven approach implemented in our prototype tool FAJITA, enable us to effectively generate test suites for container classes with rich contracts, more efficiently than other state-of-the-art tools.