We have used the High Resolution Camera (HRC-I) of the Chandra X-Ray Observatory to search for coronal (T similar to 10(6) K) emission from the archetype "noncoronal '' red giants Arcturus (alpha Bootis= HD 124897, K1 III) and Aldebaran (alpha Tauri= HD 29139, K5 III). Our program follows up previous detections of ultraviolet coronal proxies such as C IV lambda1548 (T similar to 1 x 10(5) K) and O VI lambda1031 (T similar to 3 x 10(5) K). The deep (similar to19 ks) HRC-I pointings obtained a tentative 3 sigma detection of Arcturus, with f(X) (0.2 2 keV) = 1.0(-0.8)(+1.8) x 10(-15) ergs cm(-2) s(-1) (95% confidence limits [CLs]), but failed to record Aldebaran, with an upper limit of less than or similar to 1.5 x 10(-15) ergs cm(-2) s(-1) (also at 95% CL). The corresponding L-X/L-bol ratios are a factor of ten thousand less than the Sun, a low-activity coronal dwarf. At the same time, Hubble Space Telescope Imaging Spectrograph far-ultraviolet spectra suggest the presence of a "cool absorber,'' probably near the base of the red giant chromosphere, imprinting discrete low-excitation absorptions on top of highly ionized features such as Si IV lambda1393. The hot emission zones thus are at least partially buried under a large column of chromospheric material, which would severely attenuate any soft X-rays that might be emitted. The submerged hot structures presumably are magnetic because of their high temperatures and broad C IV profiles (FWHM similar to 130 km s(-1)). Perhaps these structures are analogous to small-scale ephemeral bipolar regions seen ubiquitously on the Sun throughout the sunspot cycle and thought to be of direct convective origin. If small-scale magnetic fields indeed are present in the lower atmospheres of red giants such as Arcturus and Aldebaran, they might play a role in initiating the cool winds of such stars, perhaps through a mechanism similar to solar spicules.