No, no, not the Coca Cola!
The tarnish on real silver is different from the tarnish on silver plate. Stubborn black bits sounds like silver plate, real silver gets a fairly soft brownish colour.
Better than Coca Cola is a very hot (but not boiling) solution of 1 teaspoon cream of tartar, 1 teaspoon washing-up liquid, and a half-pint to a pint of very hot water, but don't leave it in for more than a couple of minutes. It should loosen the tarnish enough to let you finish it off nicely with a normal silver cleaner.