Provided you are a good customer (who pays their bill on time, every time), most credit card companies will lower your interest rate a bit if you just ask! Just like they will raise your spending limit if you ask.
Of course, this is not a guarantee, but one 5 minute phone call could lower your interest rate and save you thousands if you are carrying a large balance that you are trying to work out.
The interest rate on our current American Express is 9.95%. That is extremely low for credit cards. It's that low because on two separate occasions a few years ago, I called and asked them to reduce it. They did. I tried a 3rd time and they didn't, but it is currently as low as I have seen for a credit card so I am happy with it.
Once again, you stand a waaaaay better chance of this working if you actually pay your bill on time.
If you've missed a payment or had a late one, try calling after 6 straight months of paying on time.
If your nice request isn't working out, you can also try to drop the bankruptcy bomb on them. Mention that you are having trouble getting that big number down even though you are paying every month. Say you've been considering bankruptcy as an option and they'll be likely to work with you. This is because if you file for bankruptcy, they will get basically nothing. If they lower your rate and stop you from filing, they will continue to get the money that you owe them.