I've read many times that when applying for a new card you can request credit limits to be transferred from other cards.

I wasn't applying for a new card, but I had some cards with limits that were too low.

Citibank said, "We don't do that anymore," when I asked to transfer credit limit from my AAdvantage AMEX to my AAdvantage VISA.

US Bank said no as well when I attempted to move a few hundred bucks from my Personal Club Carlson card to my Business Club Carlson card.