I know I can't double money, but theoretically I think this should be possible, as a redundancy. Does anyone have experiences with that?
The Bitcoin-Qt/bitcoind client does not handle a situation where you've double spent very well at all. If you were, for instance, to spend from one client while a second was offline, then launched the second right after making the transaction then the second client wouldn't know of the transaction until either the first client re-broadcasts the transaction (perhaps not until an hour later) or else if the transaction gets included in a block. So then let's say you make another spend transaction from the second client and if it uses the same coin as was already spent then that transaction will be invalid and will never confirm.
It gets worse, ... the Bitcoin-Qt/bitcoind client doesn't have any easy way to remove that invalid transaction that you've made. So that transaction and any coins it used will be locked in a 0/unconfirmed status on the second client.
Additionally, the key pool will start to diverge between the two wallets. So your balance on one client will eventually show different from the wallet on another. And both will need to be backed up and kept separate.
So, in other words .. for most people, sharing a Bitcoin-Qt/bitcoind wallet among multiple computers ends up going badly.