We show that new low-energy photoluminescence (PL) bands can be created in the spectra of semiconducting single-walled carbon nanotubes by intense pulsed excitation. The new bands are attributed to PL from different nominally dark excitons that are "brightened" because of a defect-induced mixing of states with different parity and/or spin. Time-resolved PL studies on single nanotubes reveal a significant reduction of the bright exciton lifetime upon brightening of the dark excitons. The lowest-energy dark state has longer lifetimes and is not in thermal equilibrium with the bright state.