Ok, so Disney's CC system does indeed go down from time to time. In case anyone wants to know, the charges get routed through Florida, then on to their CC processor, then back through Florida and back to California. So there are several stages where the communications can break down.
Disney has gotten better about having some redundancy than they used to, so in some cases the charges can still be processed, they just may be delayed in hitting your bank when the system is down.
But I have been there when the system is down completely. They will still take CCs, but only if they have embossed numbers on them, as the manually run them old school at that point with manual imprints and carbon paper.
It is rare they get to that point, but I can happen, as I said this is the entire system going down because communications with Florida failed. That also means they can't scan tickets (and rely on manual methods of taking days off the ticket), can't charge for parking and some attractions run at massively reduced capacity, for example the
Disneyland Railroad can only run one train when this occurs because the signal lights that tell them when there is a train ahead of them run through Florida before coming back and lighting up properly. Yes, a major safety system runs from California to Florida and back, seems a bit poorly planned to me. Pretty much everything runs through Florida to some degree or another, but that's another story.
The bottom line is if your credit card doesn't have embossed numbers (it is becoming more and more cmon for them not to) and the system goes down, you will need cash and at that point using a CC will take substantially longer, as anything over a certain dollar amount also has to be approved over the phone.
That being said, if you have a Chase account (or are willing to pay the fees), even when the entire system goes down, you can still use the ATMs as they are run by Chase and do not go through Florida.