Let $R$ be a commutative Noetherian ring with identity and $I$ an ideal of $R$. It is shown that, if $M$ is a non-zero minimax $R$-module such that $dim\mathrm{Supp}{H}_I^i\left(M\right)\le 1$ for all $i$, then the $R$-module $H_I^i\left(M\right)$ is $I$-cominimax for all $i$. In fact, $H_I^i\left(M\right)$ is $I$-cofinite for all $i\ge 1$. Also, we prove that for a weakly Laskerian $R$-module $M$, if $R$ is local and $t$ is a non-negative integer such that $dim\mathrm{Supp}{H}_I^i\left(M\right)\le 2$ for all $i<t$, then ${\mathrm{Ext}}_R^j(R/I,H_I^i\left(M\right))$ and ${\mathrm{Hom}}_R(R/I,H_I^t\left(M\right))$ are weakly Laskerian for all $i<t$ and all $j\ge 0$. As a consequence, the set of associated primes of $H_I^i\left(M\right)$ is finite for all $i\ge 0$, whenever $dimR/I\le 2$ and...