Block intersection polynomials for a set of points \(S\) in a design provide information about the number of blocks intersecting \(S\) in a given number of points. The authors apply these polynomials to certain designs; in particular, they show a 6-\((14,7,6)\)-design and a resolvable 2-\((55,11,2)\)-design cannot have a repeated block, while every block in a 4-\((23,8,6)\)-design or a 5-\((24,0,6)\)-design can have multiplicity at most 2.