Yes, a refinancing can easily lower your annual mortgage interest by $6K per year (it did for me).
It depends on the principal balance you refinance, the change (reduction) in interest rate and how many months of the year the refinanced mortgage was in place.
Example: If you refinanced a $200,000 mortgage at 6% to a $200, 000 mortgage at 3% early in January 2016, your annual mortgage interest cost would drop by approximately $6K.
(Principal amount) X (Change in interest rate) X (Number of months with lower interest rate/12)
$200,000 X (6%-3%) X (12/12) = $6000