I did this recently with our auto insurance. Called the existing company that we've had for 2 years. Told them I was getting quotes from other companies that are 25% lower, so I'm thinking about switching, but have some questions about how that potentially affects my multi-policy discounts.
That rep immediately transferred me to someone else who was able to get me the "new customer" rate. They cancelled the old policy, set up a new policy number, and the new premium was within a few dollars of the competing quotes, down 25%.
Like many here, I played this game with the cable company before cutting the cord many years ago. It's no different with insurance. If you say the right things and they know you are serious about leaving, you get transferred to the retention dept where they have the authority to meet the competitor's new-customer price.
Retired at 52 in July 2013. On to better things...
AA: 55% stock, 15% real estate, 27% bonds, 3% cash
WR: 2.7% SI: 2 pensions, some rental income, SS later