He attended Colby Academy in nearby New London before entering Harvard College. He received a bachelor"s from Harvard in 1865, and graduated from Harvard Law School in 1868.
He then opened a law practice in Boston. He held the post of Judge Advocate of the Massachusetts Militia"s First Brigade at one point during his career. Brackett, a Republican, served on the Boston Common (City) Council he later served in the Massachusetts House of Representatives from 1877 to 1882 representing Boston from the 17th Suffolk District, 17th Suffolk District and again from 1884 to 1887.
From 1885 to 1887 Brackett was Speaker of the House.
These banks were designed to encourage thrift among the working class. From 1887 to 1890 he served as the 34th Lieutenant Governor of Massachusetts under Governor Oliver Ames.
In 1889, when Ames retired, Brackett was elected to succeed him as Governor, holding office from 1890 to 1891. During his year in office, he effectively advanced an agenda of tax reform and advocated further improvements in Massachusetts prisons.
However, he was defeated for re-election in 1890 by the Democrat William East. Russell.
Brackett then returned to his Boston law practice, remaining active in his party: in 1892 he was a delegate to the Republican National Convention and he served as a presidential elector in 1896 and 1900. In 1887 Brackett built a Queen Anne style home at 87 Pleasant Street, Arlington, Massachusetts where he lived until his death in 1918. lieutenant is listed in the National Register of Historic Places as part of the Arlington Center Historic District, and is also part of the local Pleasant Street Historic District.
In 1916, the Massachusetts legislature and electorate approved a calling of a Constitutional Convention.
The Brackett School at 66 Eastern Avenue, Arlington, Massachusetts, built in 1931, was named after him. Brackett was the executor of the estate of Abijah Ellis who was brutally murdered on Presidential Election Day (1872) and dumped in the Charles River just days prior to the Great Boston Fire of 1872.
Although Leavitt Alley was tried for the murder, he was acquitted. The murder remains unsolved.
In May 1917, Brackett was elected to serve as a member of the Massachusetts Constitutional Convention of 1917, representing Massachusetts" 8th Congressional District.