Photometric and spectroscopic data on the Type Ib SN 1984L and the probable Type lb SN 1985F near maximum light and at late times appear to lead to contradictory conclusions. At early times a low mass and/or a high explosion energy is indicated, while at late times a high mass and/or a low explosion is indicated. Several possibilities for resolving the discrepancy are considered. The ejection of approximately 50 M. of helium with a kinetic energy of approximately 2 x 10(52) ergs, little intermediate mass elements, and approximately 0.2 M. of Ni-56, might satisfy the observational constraints. The astrophysical site of such an explosion is elusive although a massive star that leaves behind a massive black hole is a possibility. Some problems that make this scenario unappealing are discussed.