Yes, this lesson has been learned dozens of times in the Bitcoin community by now. Never store significant quantities in web wallets of any kind, even if it presents as an "account" at an exchange. Move all significant quantities of coins to a local wallet and make a million secure backups of that wallet. When you want to use these coins, send them from your local wallet.
There is no reason not to have a local wallet. There have been major coin thefts from online services, even very reputable services, sometimes sinking the entire thing. Cryptocoins are worth a lot of money and people will steal them from you if you leave them vulnerable. Cloud coin storage is always vulnerable.
