We discuss the possibility of obtaining Fast Radio Bursts (FRBs) from the interior of supernovae, in particular SN 1986J. Young neutron stars are involved in many of the possible scenarios for the origin of FRBs, and it has been suggested that the high dispersion measures observed in FRBs might be produced by the ionized material in the ejecta of associated supernovae. Using VLA and VLBI measurements of the Type IIn SN 1986J, which has a central compact component not so far seen in other supernovae, we can directly observe for the first time radio signals which originate in the interior of a young (~30 yr old) supernova. We show that at age 30 yr, any FRB signal at ~1 GHz would still be largely absorbed by the ejecta. By the time the ejecta have expanded so that a 1-GHz signal would be visible, the internal dispersion measure due to the SN ejecta would be below the values typically seen for FRBs. The high dispersion measures seen for the FRBs detected so far could of course be due to propagation through the intergalactic medium provided that the FRBs are at distances much larger than that of SN 1986J, which is 10 Mpc. We conclude that if FRBs originate in Type II SNe/SNRs, they would likely not become visible till 60 ~ 200 yr after the SN explosion.