Technically true, but a given property is placed in a specific band based upon the value of the property.
The only real reason why this isn't a direct link is that the banding is not reassessed regularly.
I do know some people who successfully persuaded their local council to move their property to a lower-priced band based upon the alleged fall in property value..
The amount that a person has to pay depends on the budget of the local council and the banding of the property that the person lives in.