Can nickel bronze cutlery go in the dishwasher?

Not really. The nickel is, but "bronze" is a copper alloy. Generally, it's the reactive detergent that stains or pits the copper. You could sterilize it with the hot water after cleaning detergent residue out of your washer. Otherwise, a safe effective and inexpensive polish is a paste made from salt and vinegar. Don't get too abrasive with the salt, dissolve it in a little hot vinegar first - then dunk and rub to desired finish before rinsing and drying before restoring them. I like to rub a few drops of culinary edible oil to my cloth in the final "polish" phase before re-storage to protect the finish. Speaking of finish... Depending on the age and manufacturer, some have protective coating before anything you do would penetrate the oxidized copper in the bronze alloy mix. The great thing about salt and vinegar is NO noxious residue to consume. Best wishes! I think bronze ware is the most beautiful setting in the world. I even like it rustic and oxidated or mottled - just gorgeous, and quality pieces with a good alloy mix can last several lifetimes and counting...