For people outside of Wellington, that side of Willis street is a dedicated bus lane and no other traffic can use it. Traffic heading in that direction must use an alternative route such as Victoria Street, which runs parallel.

Sometimes you see tourists going down it by accident.  The signs leading onto it aren't that great and could easily be missed if you weren't familiar with the area. Sometimes you'll hear the buses blast their extremely loud horns in protest, scaring pedestrians.  I just find that to be overly arrogant as once you are on that stretch there are very few opportunities to get off it until you get to the end.