Building structures can be critically affected by impulsive loads such as blasts, collisions, gusts, and pulse-dominated earthquakes. The addition of nonlinear energy sinks (NESs) in buildings has been proposed as a means to rapidly and passively dissipate the energy in a system exposed to this type of loading. This rapid dissipation occurs because the essential nonlinearity of the NES allows it to resonate with any mode of the structure and engage in targeted energy transfer, the nearly one-way transfer of energy to the NES where it is locally dissipated. Additionally, the NES couples the modes of the structure and facilitates the transfer of energy from the lower modes of the structure to the higher modes, where it can be dissipated at a reduced time scale. In this study the experimental performance of a system of multiple NESs in a large 9-story test structure is discussed. Two different types of NESs are used, each of which employ a different type of restoring force; one type of NES utilizes a smooth restoring force that is roughly cubic, while the other utilizes a linear restoring force coupled with one-sided vibro-impacts. To load this system, an impulse-like ground motion is applied via a large shake table. The results of this study show that the system of NESs greatly improves the performance of the structure across a wide range of impulse amplitudes by reducing and very rapidly attenuating its response.