cddt:Correct. This would likely only require rectification if you were to do any modifications to the stormwater connection on the property that also required building consent.
Ragnor:
"Downpipes have to be connected to a stormwater system and are not allowed to discharge onto the ground surface"
I assume this is not retrospective. Our old house definitely had one downpipe discharging onto the ground... built 1930 though.
The original post itself is an interesting one. I believe that there's no explicit rule that states you have to manage your stormwater entirely within your own boundaries.
There is a by-law (a copy linked below) which would probably be worth looking at. It has a number of purposes but in this case it helps protect the public from 'nuisance' from damaged or misused stormwater systems.
My recommendation would be (as others have said) to write a nice letter explaining the situation but ensure there is a date for the work to be completed by. If the nuisance from their damaged/poorly maintained stormwater system is what you consider to be impacting you or your property then you should consult a lawyer.
FWIW i'm an Architect in Auckland.

