Place the potatoes in a large pot and cover with water. Place over high heat and cook the potatoes until just tender. Drain and let cool slightly. Peel while hot, cube, and season with the dry mustard. Cool for 5-10 minutes in the refrigerator. Add the celery, onion, sliced sweet pickles, and dill weed.
Thin the mayonnaise with vinegar and stir into the above ingredients. Season with salt and pepper, to taste. Slice the hard boiled eggs on top, if using, and season with paprika, to taste.