I have several computers at home that handle bitcoins. As the blockchain is already at least 20 GiB large, I think it'd make sense if the Qt client would support reading from a blockchain and transaction database that is located elsewhere, i.e. network accessed storage.
This way only 1 machine would hold the blockchain and databases up to date, rather than more than 1.
This exists and is called a SPV client. And SPV client, for example Multibit, or the Android Bitcoin Wallet can be pointed at a node that will store and query the blockchain for it and do verification against the UTXO set. It can also be pointed at arbitrary nodes, but using one of your own has privacy and security advantages.
There are plans for SPV support for the wallet in Bitcoin Core, eg #3882.
However when and how this will happen is not entirely clear, so I'd suggest to use an already available SPV client.
You can already do this with bitcoinqt by setting the datadir in bitcoin.conf file or setting the data directory when you install the application. That being said, I don't know if it would support multiple clients open at any given time. I run this setup myself, but only have one client open at a time.