"Neither rain, nor sleet, nor gloom of night stays these couriers from the swift completion of their appointed rounds."
that was never an official Post Office slogan. Back in the 1890's the architect who designed the New York City General Post Office included it in the building design, but the PO never claimed it.

It was originally said by the Greek historian Herodotus about 2500 years ago and referred to the Persian mounted postal couriers during the war between the Greeks and Persians about 500 B.C.