We have used the High Resolution Camera (HRC-I) of the Chandra X-Ray Observatory to search for coronal (T~106 K) emission from the archetype ``noncoronal'' red giants Arcturus (α Bootis=HD 124897, K1 III) and Aldebaran (α Tauri=HD 29139, K5 III). Our program follows up previous detections of ultraviolet coronal proxies such as C IV λ1548 (T~1×105 K) and O VI λ1031 (T~3×105 K). The deep (~19 ks) HRC-I pointings obtained a tentative 3 σ detection of Arcturus, with fX(0.2-2keV)=1.0+1.8-0.8×10-15 ergs cm-2 s-1 (95% confidence limits [CLs]), but failed to record Aldebaran, with an upper limit of <~1.5×10-15 ergs cm-2 s-1 (also at 95% CL). The corresponding LX/Lbol 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 λ1393. 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~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.