I'd say it's more normal to focus on great summer gigs, but I know plenty of people who have been able to go to school in DC and snag great internships during the semesters. I know a guy who was in American's SIS who worked part time for a consulting firm, a Congressional committee, and some think tank (I don't know the details) during the two years he was in the program.
During the summer, there are a lot of interns in DC from any school you can imagine, but the folks with the most sought after internships (high profile Senators' offices, Justice Department, State Department, etc) are typically from better schools.