Technically, yes, but there's very little actual competition on many routes. Domestically in NZ between the main centres there's Air New Zealand and Jetstar. I doubt I'll ever fly Jetstar as I know too many people who've had Jetstar flights cancelled on them at the last minute. Ten years or so ago it was a different story, with more airlines competing.
On minor routes Jetstar is bad, but I flew domestically with them half a dozen times last year.
The worst problem I experienced was a 30-delay one time.