As Bruce says, we know we shouldn't give our credit card details to somebody who has called us. The genius described in the linked article reminds us that we can't even safely give our details when we make the call.
Moving house recently, I've been interested to note the differing levels of security used by different companies.
BT, Water, Gas and Electricity were the worst offenders: if you call them and give an account number and name, they'll do whatever you tell them. There's probably not much scope to defraud someone (although I'm not smart enough to spot loopholes of this sort) but you could certainly cause mischief (closing somebody's account and forwarding their bills elsewhere could be a way of ruining their credit history)
From the article, it seems AT&T go a step further: you can call them and get them to redirect any number to anywhere else.... I wonder if you could redirect a landline to a premium rate number. It'd be a neat way to make some free money.
(Note: I'm not actually advocating anyone actually try this.... I quite like being employed and free. Being unemployed and/or in prison isn't currently on my "to do" list)