At the start of the movie, set in the early 30's Rose asks Buddy what Columbia University is and she's impressed that Mrs. Hillyer lived in New York and went there. Columbia didn't go co-ed until the 1980's, so she would have attended Barnard College (established in 1889),- one of the "Seven Sisters" - for her undergraduate degree.
shareBarnard College was affiliated with Columbia University from its founding and seems to have been referred to as part of Columbia University during the time in which the movie is set. With a little searching (thanks Google Books), I found a 1922 Columbia University announcement about Barnard College that states bluntly that "Barnard is the undergraduate college for women of Columbia University." A 1904 history of Columbia University includes the line that "Barnard College ... cares for the undergraduate women students of Columbia University."
-- TopFrog