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Weakly coupled linear beams with slight disorders have been shown to exhibit frequency veering and mode localization. Such systems have been used as the theoretical basis for the design of nanosensors, for example mass sensors and acceleration sensors. However, mode localization may generate relatively high amplitude responses in beam systems; the system dynamics is then better represented by including its nonlinear characteristics. This paper examines the transverse vibrations of a pair of slightly disordered nonlinear cantilever beams which are coupled through a weak spring. The disorder in the system may arise from slight differences in the material properties or in the geometry of the component beams. The variation in the system natural frequencies and normal modes are investigated. In particular, we explore the effects of variation of the level of disorder on the ability of the system to act as a sensor for changes in mass.