It seems to be possible to control how many results you get for a particular stop by appending the desired number of results e.g. https://www.metlink.org.nz/api/v1/StopDepartures/STOP_NUMBER/NUM_RESULTS_YOU_WANT where NUM_RESULTS_YOU_WANT is one of 20, 40, 60, 80, 100, 120
This is useful for busy stops where the bus(es) you are interested in may not be in the next 20 results (which seems to be the default).
Examples
curl https://www.metlink.org.nz/api/v1/StopDepartures/5014/120