Physicians and other health sciences professionals need continuous training, not only in technical aspects of their activity but also in nontechnical, transversal competencies with a cost-efficient impact on the proper functioning of healthcare. The objective of this paper is to analyze the behavioral change among health professionals at a large public hospital following a training intervention on a set of core nontechnical competencies: Teamwork, Adaptability-Flexibility, Commitment-Engagement, Results Orientation, and Leadership Skills for Supervisors. The 360◦ Multisource Feedback (MSF) model was applied using three sources of information: supervisors, co-workers, and the workers themselves (self-assessment). A quasi-experimental pretest–post-test single-group design with two points in time was utilized. The training intervention improved the scores of only one of the trained competencies—the “Results Orientation” competency—although the scores were slightly inflated. Moreover, significant discrepancies were detected between the three sources, with supervisors awarding the highest scores. The magnitude of behavioral change was related to certain sociodemographic and organizational variables. The study was not immune to the ceiling effect, despite control measures aimed at avoiding it. The empirical evidence suggests that the 360◦ MSF model must be maintained over time to enhance and reinforce an evaluation culture for better patient care.