Bitcoins value does indeed affect the price of all altcoins since BTC/AltCoin trading pairs are the most commonly used and is usually the standard go to exchange pair on all cryptocurrency exchanges. So can Altcoin break all the relationship from bitcoin and run as a independent currency.
Many trading platforms offer trading pairs independent of bitcoin. The more the market develops, the greater the independence will be received by alternative currencies. Everything depends on the demand in the market.