Diagnosability is an important property of discrete-event system (DES) from the perspective of diagnosis. It requires that every fault can be detected and isolated within a finite number of observations after its occurrence. In numerous literatures, diagnosability is studied under the assumption that an observation is certain, i.e., the observation corresponds to the sequence of observable events exactly taking place in the DES. But in practical applications, the assumption may become inappropriate. Due to various reasons such as the precision of sensors and noises in transmission channels, the available observation may be uncertain. This paper focuses on the diagnosability of DESs with uncertain observations. It extends the definition of diagnosability to cope with uncertain observations. Methods are given to check the diagnosability with three types of uncertain observations accordingly. In a more general scenario where multiple uncertainties exist in the observation, a method is also provided to check the diagnosability with all the uncertainties of the observation together.