If you have other card(s) with a better rate, particularly if you have a very low rate BT offer for that card, I've found that playing CCCs against each other can work. One of my favorite strategies:
"Golly, y'know, I really like your card, and I've been a good customer for years and years and years and never been late or bad in anyway, but my XYZ card has an interest rate that's 1/3 of what your card is charging me. Isn't there anything you can please do to help???" (Insert puppy eyes here, they can't see it over the phone but they'll hear it in your voice) Most of the time, I've found that they'll match the other company's rate. You have a lot more leverage if you've got a good record with them.
If Rep #1 doesn't want to help, ask for a supervisor. If they won't help, thank them, be super polite, and transfer everything you can on to the other card. Wait until it posts, then rinse and repeat, adding how you really like this card and want to be able to use it, but it's just sooooo high you can't justify it, even though it's really your favorite card... and so on and so forth...
I've used this one (okay, not verbatim...) on both Discover and Chase. Against each other. Never had to actually use the BT non-bluff, but each company probably had a good idea what the other was offering. I think my scores were in the upper 600s-lower 700s at the time; I wasn't monitoring my scores at that point.
-MsMS