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.
Altcoins are literally dependent by all means in being a crypto coin but when it comes to market prices, we can say that the price of bitcoin is affecting their market prices. I think the reason for this is trading because coins are traded to have another value of coins which is usually bitcoin. With that, if the value of bitcoin decreases, their prices decreases as well.
I really don't think that they can be entirely independent with each other because coins are traded. Maybe they could be in means of a country using that specific coin as their own currency which really sounds impossible too.