To easily get rid of really stubborn deep down tarnish, use
Tarn-X (or however it is spelled) followed by a polishing with whatever polish you typically use. By the time you polish it, it will be easy to quickly shine it up. Someone else once suggested to em to put aluminum foil on the bottom of the bath tub (shiny side up), put tarnished items flat down on it, fill the tub a little bit with Spic and Span and hot water, then cover the silver items with more aluminum foil, shiny side down this time). Apparently this causes the trnish to transfer to the foil. I never tried it, but have heard from others that it works.