Modern cyber-physical systems rely on complicated sensing pipelines to produce sensor estimates, which inherently contain uncertainty and may exhibit transient and recurring abnormalities. It is important to ensure system safety under such unavoidable uncertainty. We present a framework for CPS safety assurance under sensor uncertainty based on two key notions. First, we introduce tolerance contracts on sensor estimates, which specify how much, how long, and how frequently sensing abnormalities are permitted. Second, we leverage quantitative safety, which measures how close a CPS is to violating its safety requirements. Together, these notions enable rigorous reasoning about CPS safety in the presence of sensing abnormalities. We formalize the syntax and semantics of tolerance contracts and develop sound reasoning techniques for contracted CPSs. In particular, we develop a special invariant-style technique dedicated for reasoning with recurring abnormalities. A water tank case study demonstrates how different contract designs can be used to ensure system safety despite sensor uncertainty.