The mailman slogan

You're thinking of the classic slogan for the United States Postal Service (USPS):

"Neither snow nor rain nor heat nor gloom of night stays these couriers from the swift completion of their appointed rounds."

This phrase is often shortened to "Neither snow nor rain nor heat nor gloom..." and is often used to describe the dedication and perseverance of mail carriers, who work in all kinds of weather conditions to deliver mail to customers.