Or will they feed off of Bitcoin's carcass and take off?
In most cases the Bitcoin dump usually effects Altcoins too, most Altcoins take a dip too.
But there are a few coins that don't really feel the effect of a Bitcoin - or generally speaking a market dip
It depends on some alts use or applications some were good to be hold yet at this moment and soke since their price increases in a bit or greater than their ICO amount, btc price may affect them a little but more often it's not, unless they ised alt to buy more btc.