We measured diffractive scattering cross sections of alpha-particles on carbon, aluminium, copper and lead using the beam of the Dubna synchrophasotron at 3.64 GeV/nucleon. The data are compared with model predictions of the Glauber-Sitenko multiple scattering theory. Large discrepancies between experiment and calculations appear at four-momentum transfers above ~ 0.25 (GeV/c) 2. Various theoretical attempts to improve the description of high-energy alpha-scattering data are discussed.