Collision resilience is an important feature of robots deployed in unstructured and partially unpredictable environments. Herein, a novel dual stiffness (DS) tensegrity platform to integrate collision resilience into a robot body is proposed. The proposed DS tensegrity platform is rigid during normal robot operation, but softens upon collision to withstand the impact. The DS behavior is achieved by means of a novel DS strut that is rigid, but can buckle without breaking under high loads, thus preventing damage to the robot. Compression tests and finite element method simulations show that both the DS struts and DS tensegrities undergo substantial stiffness change with maximum load-bearing ratios up to 10.5 and 5.74, respectively, before and after buckling. These DS tensegrity structures are integrated into two types of robots, a drone and a rover, that are shown to withstand falls from 2 and 5 m, respectively. The mechanical tunability of the proposed DS tensegrity system makes it suitable for impact attenuation in a wide range of situations and robot types.