This study investigates the structural capacity of masonry arch bridges affected by geometric defects, using a rigid-block limit analysis method in which the structure is modelled as an assembly of rigid blocks. Three defect types—brick loss, surface exfoliation, and lack of mortar joints—are considered, as they are commonly observed during visual inspections. The effects of such defects are introduced as geometric modifications within the model, with random parameters sampled from uniform distributions associated with different severity levels. Monte Carlo simulations are performed to evaluate the variability in the collapse multiplier across different defects scenarios. The analysis is applied to two representative bridge configurations, with net spans of 6 m and 16 m, in order to assess the influence of span length and defect typology on collapse response. For each case, results are presented in terms of mean collapse capacity, standard deviation, and reduction relative to the undamaged condition. The study highlights how both the nature of the defect and the extent of geometric variability influence structural performance. Results also indicate that large parameter ranges may lead to significant dispersion in the response and, in some cases, to changes in the governing collapse mechanism.