Did American express have the right to raise rates?

Any businesses have a right to determine what they charge. It is then the consumer's right to decide whether to do business with these businesses based on the charge. So, if the company is charging a ridiculous rate, it will be likely that they will not have the volume of business. With credit card, it is a contract you sign with them. In the legal fine print, I am sure you will find that this right was stated and signed off by their customers.