Well now, you've got me to thinking, and lo and behold I've got my handy-dandy copy of The Elegant Universe nearby. Now, we know from Universe that there are probably a total of (drumroll please)
nine spatial dimensions!
Coincidence? Of course not. Moreover, it turns out that six of them are...curled up! Now what is it that cats do more than anything excepting, perhaps, hedgehogs and armadillos? Of course! So here we have the explanations for the nine lives - three of them are curled up in the "extended" dimensions we can see, and the other six are curled up in the curled-up dimensions we can't see.
Now, the small problem is that all of the lives are taking place concurrently, so all nine are going through the air after we push our hypothetical button. At least we have a start, though, and I must thank my esteemed colleague Wolfman.
Oh, and of course there's that pesky tenth dimension, or time, but I don't think that need detain us for the moment....