Nearly 20 years into Lost Bayou Ramblers’ career, the band can claim famous fans and Grammy nominations. The Violent Femmes’ Gordon Gano and the Pogues’ Spider Stacy jumped at the chance to collaborate with the groundbreaking Cajun band. The Ramblers’ other career highlights include high-profile TV appearances, opening for Arcade Fire and their musical contributions to the Oscar-nominated Beasts of the Southern Wild.

Lost Bayou Ramblers’ greatest feat appears most clearly in the ban...