Sorry for the very noob question, but I googled for it without finding a proper answer.
What I mean is not moving your wallet from a provider to another, but having multiple different wallets containing different amounts of bitcoins.
Well you could use different client , Bitcoin Core , Electrum , Multibit HD etc .
But I don't see why while you can use Multiple adress , I mean this why adresses exists . Even if you see "1 BTC" on your wallet and it's received on two multiple adresses , the blockchain will show the adresses balance and not the wallet ,so you are safe and annonymus .