By numerically solving the quantum kinetic equations we compute the range of parameters where the ν μ -> ν s oscillation solution to the atmospheric neutrino anomaly is consistent with a stringent big bang nucleosynthesis (BBN) bound of N B B N e f f 3.6. We show that this requires tau neutrino masses in the range m ν τ 4 eV (for δm 2 a t m = 10 - 2 . 5 eV 2 ). We discuss the implications of this scenario for hot+cold dark matter, BBN, and the anisotropy of the cosmic microwave background.