Regarding ANZ cards, the T&C's state:
Not available ... where you currently hold or have closed an ANZ Frequent Flyer, ANZ Frequent Flyer Gold, ANZ Frequent Flyer Platinum or ANZ Frequent Flyer Black credit card within the previous twelve months. This offer applies only once per customer.
Conventional wisdom is that you can get 1 of each flavour, and this has worked for me.
Question is if anyone has tested the waters and knows whether these cards are churnable?