However if thats the case, wouldnt you need to download 10s of wallets if you plan on trading at least 10+ coins? The other thing is if you do this, wouldnt you need to pay those sending fees like bitcoin to do this? So in other words, keeping a lot of btc or altcoins in your bittrex wallet is not secure? I had thought with a bittrex account, you dont even need wallets anymore etc...
You just need to look at the history of crypto exchanges (Mt. Gox, Cryptsy, Bitfinex, BTC-e) to realise that there is a significant risk holding your coins on an exchange. In general, the rule is if you don't own the private key you don't own the coin. If you trade then the exchange getting hacked, shut down or going bankrupt is an additional risk you have to take on top of the risk in trading.
Depending on what coins you own there are several multi-coin wallets available.