The late-type giants and supergiants of the "hybrid chromosphere'' class display signatures of cool (T less than or similar to 2 x 10(4) K) winds together with hot emission lines from species like C IV (T similar to 10(5) K). A survey of such stars by Reimers et al. using ROSAT reported numerous X-ray detections (T similar to 10(6) K), strengthening the (then heretical) idea that hot coronae and cool winds can coexist in luminous giants. However, several of the candidate sources were offset from the predicted stellar coordinates, calling into question the identifications. In an effort to secure better knowledge of the X-ray luminosities of the hybrids, the ROSAT fields from the Reimers et al. survey were reexamined, exploiting the USNO-A2.0 astrometric catalog to register the pointings to a few arcseconds accuracy. On the basis of positional mismatches, at least two of the previously reported detections of key hybrid stars - gamma Dra (K5 III) and beta Aqr (G0 Ib) - must be rejected. The new X-ray upper limits for these stars, combined with the remaining candidate detections (and nondetections) from the original survey, place the hybrids into the same "X-ray deficient'' category as the "noncoronal'' red giants like Arcturus (alpha Boo: K1.5 III) and Aldebaran (alpha Tau: K5 III). A few of the hybrid X-ray sources are exceptional, however. The archetype alpha TrA (K2 II- III), in particular, is securely detected in terms of positional coincidence, but its anomalous, contradictory coronal properties suggest that an unseen companion - a young hyperactive G dwarf - might dominate the X-ray emission.