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.
I started trading cryptocurrencies one year ago, and till now I think that altcoin dependence on BTC is slowly falling. It used to be alot more dependent, but as BTC dominance is falling, it gets more and more independent. So as the time will pass I think the moment will come and it should when altcoin markets wont solely depend on BTC market.