We have searched the Roentgen Satellite (ROSAT) All-Sky Survey (RASS) database at the positions of about 100 magnetic Bp-Ap stars of the helium-strong, helium-weak, silicon, and strontium-chromium subclasses. We detect X-ray sources at the positions of 10 of these stars; in four cases the X-ray emission presumably arises from an early-type companion with a radiatively driven wind, while we believe that the magnetic chemically peculiar (CP) star is the most likely X-ray source (as opposed to a binary companion) in at least three and at most five of the six remaining cases. The helium-strong stars have X-ray emission levels that are characteristic of the luminous OB stars with massive winds (log L(sub x)/L(sub bol) is about -7), whereas the He-weak and Si stars (which generally show no evidence for significant mass loss) have log L(sub x)/L(sub bol) values that can reach as high as about -6. In contrast, we find no convincing evidence that the cooler SrCrEu-type CP stars are intrinsic X-ray sources. We discuss the X-ray and radio emission properties of our sample of CP stars, and argue that both types of emission may be magnetospheric in origin; however, there is clearly not a simple one-to-one correspondence between them, since many of the magnetic stars that are detected radio sources were not detected as X-ray sources in the present survey.