There are many unknowns about how fatigue accumulates in composite structures. We know in one case that the VS mounting lugs failed laterally, all at once, across many layers of composite material, in AA587. This must have taken the structural engineers somewhat by surprise.
Composite structures gradually bought their way on the airframe over the decades, and it's true, they don't deflect much before failure.
DeSitter, ClippedCub, just in fairness to the pilots of AA587, in 2005/6 the Canadian Board and the NTSB jointly investigated two more rudder incidents on aircraft of the same vintage.
The first being the Air Transat case of a rudder oscillating violently and breaking off while cruising on autopilot, and the second a Fedex aircraft in which severe delaminations had been found - thankfully before it misbehaved.
To quote the Canadian report,
"The NTSB investigation into this occurrence has determined that the initial delamination was the result of the infiltration of hydraulic fluid in the honeycomb material of the rudder, which appears to be linked to the vulnerability of rudders built before modification 8827, affecting about 370 Airbus A300-600/A310 and 40 Airbus A330/A340. Tests of this rudder in a depressurisation chamber resulted in significant further growth in the damage. Although a direct correlation between the Air Transat and the FedEx event has not been established at this time, the event confirms that significant delamination of these rudders can progress unnoticed, in spite of the present maintenance standards in place."
Transportation Safety Board of Canada | Transportation Safety Board of Canada - Aviation Safety Recommendation A06-05
It turned out that such tap tests as had been carried out up to that time were on the wrong parts of the rudder/fin assembly.
So the good news is that the A300/A310 problem has now been found and rectified, and should not recur; and that it was not the direct result of composite construction. The other 'news' is that (since the FDRs of the time recorded pedal movement and rudder movement, but were not able to 'say' whether the pedals moved the rudder or vice versa) AA587 may well not have been pilot error at all.