Most competitive battles allow the use of 2 of the same item, although read the rules of the one you are entering, as it is occasionally banned.
All (as far as I am aware) ban the use of Pokemon that share a pokedex number. This means not only can you not use 2 of the same Pokemon, but you can't use 2 different forms of the same Pokemon on the same team as well - e.g. you can't have a Rotom Wash and a Rotom Mow on the same team, because they share a pokedex number.
Like I said, read the rules for the one you are interested in, as they do differ somewhat.