Will auto insurance companies ever give you a lower rate if you threaten to leave and go with another carrier?
Sometimes with other services, providers might give you a lower rate to stick around (think internet/cable providers, etc).
You can haggle a lot of different services, but insurance just isn't one of them. It is possible to notify your current insurance provider that you are shopping for a lower rate and they may offer a review of your policy to see if you qualify for any additional discounts or completely re-write your policy to take advantage of your most recent favorable rating factors, but there is no guarantee that this will result in a decrease to your rate. Generally speaking, you will see the most significant savings by switching to a new carrier.