By performing extensive simulations with unprecedentedly large system sizes, we unveil how rigidity influences the fracture of disordered materials. The largest damage is observed close to rigidity points when the rupture thresholds are small. Fewer bonds are broken when the thresholds are increased. Irrespectively of network and spring properties, a more brittle fracture is observed upon increasing system size, even in sub-isostatic networks where the underlying force chains govern the mechanical response. Most of the fracture descriptors show power-law size-scaling dependent on rigidity properties. Strikingly, the maximum stress drop, a proxy for brittleness, displays a universal doubly-non-monotonic dependence on system size and can be used as a novel parameter to identify different failure regimes. Our results indicate how to tune these regimes. Finally, we speculate how the size-induced brittleness is influenced by thermal fluctuations. arXiv:1907.11466v1 [cond-mat.soft]