We discuss the coronal spectra of a sample of cool stars observed with the spectrometers of the Extreme Ultraviolet Explorer (EUVE). The emission measure distributions show (a) a relatively weak component between 0.1 MK and 1 MK, (b) a dominant component somewhere between 2 MK and 10 MK, and (c) in all cases but one a component in the formal solution at temperatures exceeding ≈ 20 MK. Where this hot tail is not associated with a real hot component, it is a spurious result reflecting a lowered line-to-continuum ratio, which, for instance, may be the result of a low abundance of heavy elements or of resonant scattering in some of the strongest coronal lines. We suggest that in Procyon’s corona photons in the strongest lines formed around a few million Kelvin undergo resonant scattering in a circumstellar medium, possibly a stellar wind. The flare spectrum of AU Mic suggests that resonant scattering may also occur in dense, hot flare plasmas. The electron densities of the 5–15 MK component are some three orders of magnitude higher than typical of the solar-like component around 2 MK; the volume filling factors of the hot components are therefore expected to be relatively small.