We report on the experimental observation and theoretical study of the bound state resonances in fast atom diffraction at surfaces. In our studies, the 4He atom beam has been scattered from a high-quality LiF(001) surface at very small grazing incidence angles. In this regime, the reciprocal lattice vector exchange with the surface allows transient trapping of the 0.3-0.5 keV projectiles into the quasistationary states bound by the attractive atom-surface potential well which is only 10 meV deep. Analysis of the linewidths of the calculated and measured resonances reveals that prior to their release, the trapped projectiles preserve their coherence over travel distances along the surface as large as 0.2 μm, while being in average only at some angstroms in front of the last atomic plane.