Nebraska has a U.S. Senate race this year. Republican incumbent Deb Fischer is running for re-election. So far no Democrat has announced, but an independent candidate, Dan Osborn, has announced. The Democratic Party might endorse him. A recent poll showed the two virtually tied. See this story.
In 1936 Nebraska elected an independent U.S. Senator, George Norris. Norris had first been elcted as a Republican but he switched to being an indepedent in 1936, and he got re-elected. But in 1942, when he again ran as an independent, he was defeated.