The reliability and safety of advanced driver assistance systems and autonomous vehicles are highly dependent on the accuracy of automotive sensors such as radar, lidar, and camera. However, these sensors can be misaligned compared to the initial installation state due to external shocks, and it can cause deterioration of their performance. In the case of the radar sensor, when the mounting angle is distorted and the sensor tilt toward the ground or sky, the sensing performance deteriorates significantly. Therefore, to guarantee stable detection performance of the sensors and driver safety, a method for determining the misalignment of these sensors is required. In this paper, we propose a method for estimating the vertical tilt angle of the radar sensor using a deep neural network (DNN) classifier. Using the proposed method, the mounting state of the radar can be easily estimated without physically removing the bumper. First, to identify the characteristics of the received signal according to the radar misalignment states, radar data are obtained at various tilt angles and distances. Then, we extract range profiles from the received signals and design a DNN-based estimator using the profiles as input. The proposed angle estimator determines the tilt angle of the radar sensor regardless of the measured distance. The average estimation accuracy of the proposed DNN-based classifier is over 99.08%. Therefore, through the proposed method of indirectly determining the radar misalignment, maintenance of the vehicle radar sensor can be easily performed.