No "normally" about it. The churning rules vary tremendously from bank to bank, and even from card to card, and can even vary from card offer to card offer.

At Citi, for example, they don't care whether you've cancelled or converted or still have the card, they care how long it's been since you last applied successfully for a card in the same category (such as AA personal). And with some Citi cards (Hilton HHonors 50k), you don't have to wait more than month or two before applying again. But at Citi, it can sometimes be different from one card offer to another card offer for the very same card!

At Amex (the bank), you generally have to wait at least a year since you cancelled the card.

At Chase, some people can never get the bonus for the exact same card again, others may be able to do it after a few years, YMMV. But at Chase the main trick is the word "exact", since periodically they come out with new cards, and those aren't subject to the "exact same card" rule.

So are you can see with just these three examples, it all varies widely. And it's still different at other banks, I just don't know the general rules for the other banks.