There are two risks to fly-by-wire: (1) the computers may fail, causing a total loss of control; and (2) the computer software—either by design or by accident—may not allow a pilot to do something that he has to do in an emergency to save the flight.
The first issue is addressed by having as much redundancy as practical in the computer systems and their mechanical actuators, in order to reduce the likelihood of a total failure that would make control of the aircraft impossible. Ideally, the redundancy should be great enough, and the probability of failure low enough, to at least match or exceed the expected reliability of purely mechanical systems. Fly-by-wire is still relatively young, and accidents are very rare among airliners, so the exact relative safety levels of fly-by-wire versus purely mechanical systems are still an open question.
The second problem is more ominous. On the one hand, engineers insist that pilots should not be allowed to do certain things, no matter what the circumstances, because they might damage the airplane. These engineers program fly-by-wire systems to simply refuse any control movements by the pilots that look "dangerous" to the computers. On the other hand, pilots insist that they be allowed to do anything, just in case—that's what they are there for, after all. Pilots want the fly-by-wire system to do whatever it's told in an emergency, even if it might bend the airplane. I tend to side with the pilots. There's not much point in having pilots to handle an emergency if they try to make the plane do something that will save the flight and the fly-by-wire computers simply say "no."
The problem of software bugs is also quite a concern. Although aviation software is much more carefully designed and tested than the software on your desktop computer, it is not possible to guarantee that it is bug-free with 100% certainty, and digital systems (those run by computer, such as fly-by-wire) have a nasty tendency to fail catastrophically when things go wrong. It has traditionally been a nightmare scenario of science-fiction to imagine a computer that has total control of a situation and then goes haywire, but unfortunately that scenario does have some basis in fact. When you give computers total control, you're putting your lives in the hands of whoever programmed the computers. Computers themselves are reliable, but the software that makes them behave (which is written by human beings) often is not.
The Boeing/Airbus difference is that Boeing tends to take the pilot's point of view, whereas Airbus has long taken the engineer's point of view. An Airbus is great as long as no maneuvers are required that violate the engineers' idea of what is right; if the pilot needs to do something that the computers don't like, though, he's doomed. A Boeing is great as long as you have a competent pilot who will let the computers do their job and will heed any warnings given by the computers; but if the pilot does something really stupid, the computers are less likely to stand in his way.
And yes, there have been cases where fly-by-wire might have been a problem. The Gimli Glider, a Boeing airliner that had to glide to a landing, might have crashed had the pilots not been able to perform a slip to reach the runway. Boeings let the pilots cross the controls in the way required to perform this maneuver, but an Airbus normally considers such a use of the controls to be an error, and may refuse to do it. (How true this is for Airbus, I don't know—they do have "direct law" and other features that might allow pilots to get past this—but one of the pilots claimed that they would have crashed in an Airbus, as I recall.)