Aluminum-27 nuclear spin-lattice relaxation times T t have been measured in the normal and superconducting states of dilute alloys of Mn, Cr, and V in aluminum. These measurements prove the spectrum of thermally excited quasiparticle states in the superconductor and its modification by nonmagnetic resonant impurities. Theoretical studies of such systems indicate that in the Hartree-Fock (HF) approximation the BCS character of the host is basically retained in the alloy, with a reduced pairing interaction and a small additional broadening of quasiparticle states at the gap edge. Spin-fluctuation improvements on the HF approximation renormalize the parameters involved, but do not change the basic picture. Our TI data indicate gap edge broadening an order of magnitude greater than predicted. The discrepancy is nearly as large for V and Cr impurities as for Mn, which makes it unlikely that localized spin fluctuations play an important role in the broadening. Recent speculation on the effect of dislocation pinning by resonant impurities may be relevant to these results.