We present spectra of two aging supernovae, SN 1986J in NGC 891 and SN 1980K in NGC 6946. SN 1986J was observed in 1986 and 1989, approximately 4 and approximately 7 yr after the explosion, as inferred from radio observations; SN 1980K was observed 1989, 9 years after maximum light. SN 1986J shows two components of emission: narrow lines (DELTA-upsilon < 600 km s-1) of H, He, N, and Fe, and broad lines (DELTA-upsilon > 1000 km s-1) of [O I], [O II], and [O III]. The flux in the narrow H-alpha has declined by a factor of 10 from 1984 to 1989, while the flux in the broad oxygen lines has not decreased from 1986 to 1989. We attribute these two components to two different emission sites-a circumstellar shell for the narrow lines (already suggested by the very powerful radio emission from SN 1986J) and the actual stellar interior (without hydrogen) for the broad lines. We have developed diagnostics for the oxygen line ratios to estimate the density in the debris as 6 x 10(8) < n(OI) < 2 x 10(9) at a temperature of 3000 < T < 4000 K. We find that the oxygen debris is highly clumped, filling only 0.001-0.003 of the volume and that the mass of warm neutral oxygen is 0.1 < M(OI) < 0.3 M(.). Overall, the properties of SN 1986J are consistent with the explosion of a massive star with an extensive mass loss. SN 1980K, which was observed at maximum to be SN II, continues to emit in very broad lines (DELTA-upsilon > 2500 km s-1) of [O I], [O III], [Fe II], and H-alpha. Comparison with earlier observations suggests that SN 1980K halted its exponential decline in flux in the early 1980s and is now emitting at a constant rate of about 6 x 10(37) ergs s-1 in the observed lines. For both SN 1986J and SN 1980K, the underlying energy source sustaining the emission remains uncertain, but continued measurements of the ionization, temperature, and flux may help to distinguish among shocks, radioactivity, pulsar power, or accretion onto a neutron star.