In the near-Earth environment, strong bulk plasma accelerations are frequently taken to be the diagnostic of the occurrence of magnetic reconnection. In this letter, we report new and unambiguous spacecraft observations and corresponding magnetohydrodynamic (MHD) simulation of strong bulk plasma acceleration in the terrestrial magnetosheath during low Alfvén Mach number solar wind conditions, which is demonstrably not associated with magnetic reconnection. We illustrate this effect with Cluster spacecraft data that show plasma accelerations up to speeds of 1040 km/s, while the ambient solar wind speed is only 650 km/s (i.e., in excess by 60%). Based on a comparison with global MHD simulations of the magnetosphere, we show that the acceleration results from enhanced magnetic forces exerted on the plasma by “stiff” magnetic flux tubes in a low-β magnetosheath that result from the low Alfvén Mach number solar wind. The MHD simulations demonstrate that the acceleration is asymmetric, as well as the magnetopause shape, and is the result of both magnetic pressure gradient and tension forces, showing that this effect is not a simple analogy to a “slingshot effect” for which magnetic tension would dominate. Like magnetic reconnection, this mechanism is capable of producing strong plasma acceleration in the near-Earth's environment. The low Alfvén Mach number solar wind condition leading to this mechanism is often characteristic of coronal mass ejections (CMEs).