AbstractQuasi-residual designs are balanced incomplete block designs having the parameters of a residual BIBD. For λ = 1 or 2 all quasi-residual designs are also residual designs, but a single counterexample due to Bhattacharya for the design (16, 24, 9, 6, 3) shows this not to be the case for λ = 3. We examine the block structure of this type of design, and use this information to construct eight new solutions for (16, 24, 9, 6, 3); along with the Bhattacharya design, these are the only known counterexamples for λ = 3.The more general case with λ > 3 is mentioned briefly.