Majinder [3] obtained an upper bound for the number of disjoint blocks in BIB designs. In this paper we give an upper bound for the number of disjoint blocks in (i) Semi-regular GD designs, (ii) PBIB designs with two associate classes having triangular association scheme, (iii) PBIB designs with two associate classes having $L_2$ association scheme, and (iv) PBIB designs with three associate classes having rectangular association scheme. The main tools used to establish the results of this paper are the theorems proved by (i) Bose and Connor [1], (ii) Raghavarao [4] and (iii) Vartak [6].