A process that may limit the nonlinear steepening of hydromagnetic waves in a collisionless plasma is considered. The assumption that Landau damping dissipates the higher &kgr; Fourier components of a nonlinear wave more strongly than the lower &kgr; components implies that Landau damping would keep the thickness of the wave finite. A crude mathematical model of this process gives this limiting thickness Δ as Δ~ (L/2&pgr;)(B/ΔB) exp (-1/&bgr;p), where ΔB and B are the magnetic amplitude and mean magnetic field of the wave, &bgr;p is the ratio of proton pressure to magnetic pressure, and L is the initial thickness of the pulse, provided that Δ is greater than or approximately equal to the proton Larmor radius. Two structures that are qualitatively shocklike but thicker than expected from theory are used as examples to test our model. It is found that both events are in the process of steepening and their limiting thicknesses due to Landau damping are greater than the corresponding proton Larmor radius for both structures as observed at Mariner 5 (nearer the sun than 1 AU) but are comparable to the proton Larmor radius for Explorer (near 1 AU) observations. We suggest that many broad but otherwise shocklike structures may be interpreted as Landau damping limited structures.