Biden was knocked out of contention for the Democratic Nomination in 1988 because he copied a speech delivered by British Politician a year earlier (and even went so far as to fudge his own familly history a bit to make the copied lines work)
That hardly seems like plagiarism, more paraphrasing and even then it isn't like he took the entire speech - just a basic idea and two sentences (which he apparently cited everywhere except once).
"He went on to receive his Juris Doctor from Syracuse University's College of Law in 1968,[19] where by his own description he found it to be "the biggest bore in the world" and pulled many all-nighters to get by.[15][20] During his first year there, he was accused of having plagiarized 5 of 15 pages of a law review article. Biden said it was inadvertent due to his not knowing the proper rules of citation, and he was permitted to retake the course after receiving a grade of F, which was subsequently dropped from his record.[20] He was admitted to the Delaware Bar in 1969.[19]"