We compare popular school choice mechanisms in terms of children's access to better schools (ABS) than their catchment area school, in districts with school stratification and where priority is given for residence in the catchment area of the school. In a large market model with two good schools and one bad school, we calculate worst-case and best-case bounds of the Boston Mechanism (BM). We find that both BM and DA convey a non-negligible risk that catchment area priority fully determines the final assignment regardless parents' preferences. Top-Trading Cycles is an alternative that provides more access to better schools than DA.