From History of New Bedford. The New Bedford High School was established in 1827, with John F. Emerson principal, but was abolished in 1829. For eight years Mr. Emerson then conducted a private High School. In 1837 the Public High School was revivified under a mandatory state law, with Mr. Emerson at the head.
No mention of when the building in the postcard was constructed, but from its design, I'd guess the late 19th/early 20th century.
And from the information presented in a Wikipedia article*, one might assume the school is primarily a youth sports factory, with little to do with academics. (*Another piece of evidence as to why Wikipedia is a perversion of the idea of an encyclopedia.)