We report that freestanding films of vertically aligned carbon nanotubes exhibit super-compressible foamlike behavior. Under compression, the nanotubes collectively form zigzag buckles that can fully unfold to their original length upon load release. Compared with conventional low-density flexible foams, the nanotube films show much higher compressive strength, recovery rate, and sag factor, and the open-cell nature of the nanotube arrays gives excellent breathability. The nanotube films present a class of open-cell foam structures, consisting of well-arranged one-dimensional units (nanotube struts). The lightweight, highly resilient nanotube films may be useful as compliant and energy-absorbing coatings.