We report Si, C, N, Mg-Al, Ca, and Ti isotopic data for three micron-sized presolar SiC grains from the Murchison meteorite. These grains have very low C-12/C-13 and N-14/N-15 ratios, similar to grains for which an origin in classical ONe novae has been previously ascribed. Isotopic signatures in one grain (Si-28, Ti-49, and Ca-44 excesses and a very high inferred Al-26/Al-27 ratio) indicate that it in fact formed in a supernova, not a nova. Similarly, a large Ti-47 excess in another grain argues against a nova origin. The new data raise the question whether all grains previously attributed to novae might have in fact originated in Type II supernovae. The results also point to coupled synthesis of C-13 and N-15 in Type II supernovae.