SAN FRANCISCO—Federal Circuit nominee Kara Stoll has a lot going for her: a successful patent appellate practice at an Am Law 100 firm, strong connections in Washington and the patent bar, and a reputation for playing well with others—a valuable asset for a court with a history of factiousness.

The challenge facing Stoll—President Barack Obama’s latest nominee to the U.S. Court of Appeals for the Federal Circuit—is a GOP-controlled Senate that won’t be stepping lively to confirm circuit court picks, no matter how agreeable. Even in less polarized times, Federal Circuit nominees, with no home-state senator to champion them, have a history of languishing. Judge Richard Taranto waited 16 months for his 2013 confirmation vote, even though it went 91-0 in his favor. Judge Timothy Dyk, nominated by Bill Clinton in 1998, waited more than two years.