We use Heegaard splittings to give a criterion for a tunnel number one knot manifold to be nonfibered and to have large cyclic covers. We also show that a knot manifold satisfying the criterion admits infinitely many virtually Haken Dehn fillings. Using a computer, we apply this criterion to the 2 generator, nonfibered knot manifolds in the cusped Snappea census. For each such manifold M, we compute a number c(M), such that, for any n>c(M), the n-fold cyclic cover of M is large.

The second author was partially supported by NSF grant DMS 0306062. The third author was partially supported by NSF grant DMS 0204428.