Summary: Let \(R\) be a commutative Noetherian ring, and let \(\underline{a}, \underline{b}\) be two ideals of \(R\) such that \(R/(\underline{a}+\underline{b})\) is Artinian. Let \(M\) and \(N\) be two finitely generated \(R\)-modules. We prove that \(H_{\underline{b}}^j(H_{\underline{a}}^t(M,N))\) is Artinian for \(j=0,1\), where \(t=\mathrm{inf}\{i\in\mathbb{N}_0: H_{\underline{a}}^i(M,N)\,\,\text{is not finitely generated}\,\,\}\). Also, we prove that if \(\mathrm{dim}\,\mathrm{Supp}\,(H_{fa}^i(M,N))\leq 2\), then \(H_{\underline{b}}^1(H_{\underline{a}}^i(M,N))\) is Artinian for all \(i\). Moreover, we show that if \(\mathrm{dim}\,N=d\), then \(H_{\underline{b}}^j(H_{\underline{a}}^{d-1}(N))\) is Artinian for all \(j\geq 1\).