Nespresso says if you’re going to sell coffee pods, you can’t explicitly tell consumers they fit into Nespresso machines—or at least not too loudly.

The coffee machine maker alleges that Jones Brothers Coffee Co. Distribution LLC’s marketing of “Nespresso-compatible” capsules misleads consumers into believing Nespresso endorses their product. Jones Brothers “has built an entire business by unlawfully trading off Nespresso’s valuable assets and goodwill,” Nespresso said in an April trademark infringement complaint in the U.S. District Court for the Southern District of New York.

The lawsuit could test the boundary between simply using a trademark to inform customers, which is...