I have a Bitcoin wallet through an exchange, can I have a wallet not attached to an exchange? or multiple wallets? if the exchange has problems = I have problems 
There are lots of Bitcoin wallets, and I dont even rely on one. I keep my funds on different wallets. Most importantly, the Bitcoin.info wallet. I have also tried out different wallets like Xapo; which Im still using, Coinbase; stopped using it due to difficulty while logginin.Nano S ledger and Trezor are trying too; I prefer Nano S ledger.