Abstract. As Be stars are restricted to luminosity classes III-V, but early B-type stars are believed to evolve into supergiants, it is to be expected that the Be phenomenon disappears at some point in the evolution of a moderately massive star, before it reaches the supergiant phase. As a first stage in an attempt to determine the physical reasons of this cessation, a search of the literature has provided a number of candidates to be Be stars with luminosity classes Ib or II. Spectroscopy has been obtained for candidates in a number of open clusters and associations, as well as several other bright stars in those clusters. Among the objects observed, HD 207329 is the best candidate to be a high-luminosity Be star, as it appears like a fast-rotating supergiant with double-peaked emission lines. The lines of HD 229059, in Berkeley 87, also appear morphologically similar to those of Be stars, but there are reasons to suspect that this object is an interacting binary. At slightly lower luminosities, LS I +56• 92 (B4 II) and HD 333452 (O9 II), also appear as intrinsically luminous Be stars. Two Be stars in NGC 6913, HD 229221 and HD 229239, appear to have rather higher intrinsic magnitudes than their spectral type (B0.2 III in both cases) would indicate, being as luminous as luminosity class II objects in the same cluster. HD 344863, in NGC 6823, is also a rather early Be star of moderately high luminosity. The search shows that, though high-luminosity Be stars do exist, they are scarce and, perhaps surprisingly, tend to have early spectral types.