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Long paleoseismic records on mature faults suggest potentially chaotic recurrence patterns with cycles of strain accumulation and release that challenge simple slip-or time-predictable recurrence models. In apparent contradiction, the relatively small variability of earthquake recurrence times on these faults is often characterized as quasi-periodic, implying much regularity in the underlying mechanics. To reconcile these observations, we simulate one of the longest paleoearthquake records – the 24-event record from the Hokuri Creek site on the Alpine fault in New Zealand – using a physical model of rate-and state-dependent friction. In a parameter space formed by three non-dimensional parameters, a sea of parameters produces periodic earthquake recurrence behavior. Only a few models are characterized by fundamentally aperiodic recurrence patterns, in parametric islands of chaos. Complex models that produce partial and full ruptures of the Alpine fault can explain the earthquake recurrence behavior of the Alpine fault, reproducing up to 11 consecutive events of the Hokuri Creek paleoseismic record within uncertainties. The breakdown of the slip-and time-predictable recurrence patterns occurs for faults that are much longer than the characteristic nucleation size. The quasi-periodicity of seismic cycles is compatible with the nonlinear and potentially chaotic underlying mechanical system, posing an inherent challenge to long-term earthquake prediction.