In this paper, investigations into the nonlinear asymmetric vibrations of a pressure sensor diaphragm under initial tension are presented. A comprehensive mechanics model based on a plate with in-plane tension is presented and the effect of cubic nonlinearity is studied. Specifically, the nonlinear asymmetric response is investigated when the excitation frequency is close to the natural frequency of an asymmetric mode of the plate. The obtained results show that in the presence of an internal resonance, depending on the initial tension, the response can have not only the form of a standing wave but also the form of a traveling wave. In addition, damping can be used to reduce the nonlinear effect and avoid the nonlinear interactions. The results of this work will benefit the design of diaphragm-type structures used in microscale sensors including pressure sensors.