No. Some wallets (like Electrum) use an external blockchain (IIRC, Electrum hosts their own). Your client simply connects to their servers to verify things and get updates. This is good for a lot of people, because the blockchain is huge and constantly getting bigger. However it's somewhat less secure, you have to trust an external party. By running a "full node" (downloading the whole blockchain with bitcoin-qt), you are not only making yourself more secure, but also improving the security of the network (you have the full chain, so your client can help call out cheaters, to simplify things a bit). Whether this is worth the extra computing time and 6gb of data is your call.

(You can speed up the download some by downloading most of the blockchain from a torrent and copying it into your bitcoin folder. Your client will then only have to sync the blocks that happened after the torrent upload).