I found out that on larger amounts of money, CD Rates can be negotiated. They have a margin they can work with. They don't tell you this.
I had two CD's coming due out of state and on the due date got the new renewal rate and shopped it locally. Negotiated with a local institution, then called the other institution back while in their office telling them I was going to redeem the CD unless they could meet the rate I could get locally. They did say they could increase the quoted renewal rate, but they could not match the rate quote I got, they have a certain margin to work with.
When CD's are coming due, check all institutions and keep ads. When talking with the bank representative for their rates, then you can ask if they can match other ads. Sometimes they have to check and get back with you, and had them call me back and say they could give me a little higher rate than what I asked if they could match!