(Assuming that the system is the same as that in use in 2005)
If one of the two rides allows pulling another fastpass right away then you can hit both in quick succession in either order, provided you don't already hold unmatured fastpasses from elsewhere in the park. More correctly, each ride has its own "lockout" time before you can get its fastpass or, after that, yet another fastpass.
So the one that is "disconnected" is not affected by fastpasses fetched elsewhere in the park and is not affected by the other.
But when you get a fastpass from the "disconnected" kiosk, the time for next fastpass as printed will not be "sooner" than as printed on any other fastpass you already hold. This means that if Dumbo was "disconnected", the fastpass will not say you can get another fastpass "immediately." This means that the time for next fastpass, as shown, could in some cases be long after the return time.
If the kiosk is really disconnected, the time for next fastpass should be omitted.
The lockout time can be adjusted, possibly several times a day. This makes it possible for kiosks to seem disconnected some of the time and seem connected some of the time.