Originally Posted by ArizonaGuy

It doesn't. It's a common "warning" made by agents who are trying to retain a customer. Poor training or shady practice, who knows.

You'll lose any miles earned since the most recent statement closing date, typically. That's normal across the industry. I surmise this is what the agents really mean but they either don't understand the difference or for some reason it just gets explained this way.

Chase is the one who has been known to pull bonus miles for canceling less than 6 months after opening the credit account but even that's never a certainty.