High gradient radiofrequency (RF) accelerating cavities are one of the main research lines in the development of compact linear accelerators. A particular focus of these structures is for medical hadron therapy applications. However, the operation of such cavities is currently limited by nonlinear electromagnetic effects that are intensified at high electric fields, such as dark currents and RF breakdowns. A new normal-conducting High Gradient S-band Backward Travelling Wave accelerating cavity for medical application (v=0.38c) was designed and constructed by the TERA Foundation in collaboration with CERN. This cavity is being tested at the IFIC High-Gradient (HG) Radio Frequency (RF) laboratory. The main goal of the tests is understanding which is the maximum achievable accelerating gradient of this new design and characterize the dark current and breakdown formation in the structure, which could limit the applicability of this technology. In this work, we present experimental measurements and simulation results characterizing the nonlinear effects of this new accelerating cavity and first conclusions about its applicability are discussed.