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.
It could be possible if crypto will still get strong for many years and get stabilized market. But for now, altcoins still depending on bitcoin trend in the market as they have a strong tie up at this time.