It is of great significance to enhance the mechanical properties in single grain YBCO bulk superconductors, as mechanical failure limits the suitability of these materials for high-field applications. The influence of the microstructure on the mechanical properties of YBCO bulk materials was studied in 11 partially oxygenated YBCO, 11 oxygenated YBCO, and 10 oxygenated YBCO(Ag) melt-processed bulk single grains in a statistically significant way using Brazilian testing. Surprisingly, the oxygenation cracks evolving during the tetragonal to orthorhombic phase transition do not influence the average tensile strength of YBCO bulk single grains, with compaction cracks and pores as the main microstructural flaws responsible for mechanical failure. The mechanical properties of bulk YBCO single grains even improve following oxygenation, as indicated by an increase in Weibull modulus. An addition of 10 wt% Ag₂O to YBCO increases the average tensile strength and the Weibull modulus further, making the material mechanically more robust. However, an overlap in the measured tensile strengths of oxygenated YBCO and oxygenated YBCO(Ag) bulk materials reveals that not all silver-containing samples have better mechanical properties compared to oxygenated YBCO bulk samples.