Magnetic fluctuations in the solar wind show an f-5/3 power-law spectrum below the Doppler-shifted proton cyclotron frequency but steepen to f-s with s≥3 for higher frequencies. The origin of this steepening, however, remains unclear. The purpose of this study is to evaluate critically the often-employed assumption that the steepening is caused by dissipation via kinetic wave damping. For both Alfv¿n and magnetosonic waves and for a broad range of propagation angles, we show that the wave damping rate usually increases very strongly with wavenumber k. Consequently, the wave energy transfer always becomes slower than the damping at sufficiently high k, resulting in a strong cutoff in the power spectra rather than a steepened power-law. This result suggests that collisionless dissipation can not be the only physical basis for explaining the steepening. Furthermore, it casts serious doubts on the basic approach of treating magnetic fluctuations as an ensemble of linear waves. ¿ 2001 American Geophysical Union