Nonlinear effects in microresonators are efficient building blocks for all-optical computing and telecom systems. With the latest advances in microfabrication, coupled microresonators are used in a rapidly growing number of applications. In this work, we investigate the coupling between twin-resonators in the presence of Kerr nonlinearity. We use an experimental setup with controllable coupling between two high- Q resonators and discuss the effects caused by the simultaneous presence of linear and nonlinear coupling between the optical fields. Linear-coupling-induced mode splitting is observed at low input powers, with the controllable coupling leading to a tunable mode splitting. At high input powers, the hybridized resonances show spontaneous symmetry breaking (SSB) effects, in which the optical power is unevenly distributed between the resonators. Our experimental results are supported by a detailed theoretical model of nonlinear twin-resonators. With the recent interest in coupled resonator systems for neuromorphic computing, quantum systems, and optical frequency comb generation, our work provides important insights into the behavior of these systems at high circulating powers.