As SLF, I’d like to know that if/when the computer throws up its metaphorical hands, the pilot in command has the training/experience/skills to take control of the aircraft and fly it in what has often been referred to in this forum as “stick-and-rudder” mode.
I am also a SLF (whatever that stands for - assume it means "non-pilot type"). From what I understand the lowest level of Airbus control (excluding the "stuff-has-hit-the-fan-all-the-chips-are-down mechanical pitch trim and rudder) is direct law - the control surface deflection is directly proportional to the stick (or rudder pedal) position. In this regard it is exactly like a normal aircraft, and the computers merely translate the stick's position into control surface deflection without adding their own inputs or limits. In this case, and if my understanding is correct, the pilot should be able to handle the aircraft - after all, he or she has probably spent much time in his or her early flying lifetime flying airplanes that behave in exactly this way. Ultimately the Airbus aircraft are aerodynamically conventional, so they should be stable and well within a pilot's ability to fly, even without computer compensation.
I hope I did not mangle the facts too badly there, and if I did I apologize and I would welcome any corrections - but as far as I am aware, the answer to your question is "yes, they do".