The statistics for the distribution of laminar phases in type-III intermittency is examined for a symmetric bimodal map. Due to a strongly nonuniform reinjection process, characteristic deviations from normal statistics are observed. There is an enhancement of relatively long laminar phases followed by an abrupt cut-off of laminar phases beyond a certain length. The paper also examines the bifurcation structure of two symmetrically coupled maps, each displaying a subcritical period-doubling bifurcation. The conditions for a pair of such maps to exhibit type-II intermittency are worked out.