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.
Maybe, of course, only if bitcoin becomes more popular. Otherwise, there will always be a link to bitcoin, which is more reliable and stable.
Because almost all altcoins are priced with satoshi, the impact of bitcoin on the altcoin is unlikely. But I think that as the cryptocurrency market grows stronger and altcoin has its own community it can lessen the impact of bitcoin on the value of altcoin.