If a Chicagoan tells you he went to “Northeastern,” chances are he means Northeastern Illinois University, not Northeastern University in Boston. Northeastern Illinois is on Chicago’s North Side, which is indeed in northeastern Illinois—but not exactly in the Northeast (i.e., northeastern U.S.), as Northeastern U is.
Chicago’s Northeastern is about 7 miles southwest of Northwestern, in suburban Evanston—which of course is not in the Northwest as we now know it, but is so named because it’s in the old Northwest Territory, which now comprises the Midwestern states of Ohio, Michigan, Indiana, Illinois, Wisconsin, and the easternmost portions of Minnesota. They didn’t bother to change the name to Midwestern University when due to westward expansion the old Northwest became the Midwest. Probably wise. “Midwestern University” probably wouldn’t have the same kind of appeal to Coastal types.