An assembly is provided for a turbine engine. This assembly includes a rotating structure rotatable about an axis, a stationary structure and a bearing. The stationary structure forms a bearing compartment, a buffer cavity, a low pressure cavity and a high pressure cavity with the rotating structure. The buffer cavity surrounds the bearing compartment. The low pressure cavity surrounds the buffer cavity. The high pressure cavity surrounds the low pressure cavity. The bearing is within the bearing compartment. The bearing rotatably couples the rotating structure to the stationary structure.