Burlington Branch, Conn?

Where was this BURLINGTON BRANCH / CONN. used? There was never a post office in Connecticut named Burlington Branch. There was a Burlington, from 1808 to 1907, but it was not a branch. Later, a Burlington rural branch and more recently, regular branch, of Unionville opened in 1968, but that is unrelated to this 1873 marking.
Burlington did have a railroad station, on the branch line to Collinsville (and after 1876 to New Hartford) of the New Haven & Northampton Railroad. Would the branch have been called the Burlington branch if the terminus was Collinsville?
The destination, Brooks Vale, was on the main line of the New Haven & Northampton. Could this have been a route agent's datestamp?
No markings at all on the back.
Comments would be most welcome!