I think the exception to this is if you're underwater on your mortgage. Then the new bank won't give you a new loan against the house for more than its worth.
This is of course stupid on the part of everyone involved
but especially the person who currently owns your loan. Lowering the interest rate for a person with an underwater mortgage can only increase the security of it as a financial asset. If they default it's pure loss. But a new bank doesn't want to take on an underwater mortgage and even at the institution that owns your loan the left hand originating mortgages doesn't know what the right hand is doing servicing them.
This is of course stupid on the part of everyone involved but especially the person who currently owns your loan. Lowering the interest rate for a person with an underwater mortgage can only increase the security of it as a financial asset. If they default it's pure loss. But a new bank doesn't want to take on an underwater mortgage and even at the institution that owns your loan the left hand originating mortgages doesn't know what the right hand is doing servicing them.