Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$ and let $I$ be an $\mathfrak{m}$-primary ideal. Let $G$ be the associated graded ring of $A$ with respect to $I$ and let $\mathcal R = A[It,t^{-1}]$ be the extended Rees ring of $A$ with respect to $I$. Notice $t^{-1}$ is a non-zerononzero divisor on $\mathcal R$ and $\mathcal R/t^{-1}\mathcal R = G$. So, we have Bockstein operators $\beta^{i} \colon H^{i}_{G_+}(-1) \rightarrow H^{i+1}_{G_+}(G)$ for $i \geq 0$. Since $\beta^{i+1}(+1)\circ \beta^{i} = 0$$, we have Bockstein cohomology modules $BH^{i}(G)$ for $i = 0,\dots,d$. In this paper, we show that certain natural conditions on $I$ implies vanishing of some Bockstein cohomology modules.