Let $E$ bea~Banach function space over a~complete finite measure space \ \mbox{$(\mit\Omega, \mit\Sigma, \mu)$}, $E^\prime$-the K\"othe dual of $E$ and let $X$ be a~Banach space, $X^*$-the topological dual of $X$. We give a~criterion for relative sequential \mbox{$\sigma(E(X), E^\prime(X^*))$}-compactness in K\"othe-Bochner spaces $E(X)$. We characterize Banach spaces $X$ having the Radon-Nikodym Property in terms of relatively sequentially \ \mbox{$\sigma(E(X), E^\prime(X^*))$}-compact subsets of $E(X)$. Moreover, we show that $E(X)$ \ is sequentially \ \mbox{$\sigma(E(X), E^\prime(X^*))$}-complete if and only if $E$ is sequentially \ \mbox{$\sigma(E, E^\prime)$}-complete, $X$ has the Radon-Nikodym Property (with respect to $\mu$) and $X$ is sequentially weakly complete. We generalize F.~Bomball, J.~Batt and W.~Hiermayer's results concerning weak compactness and sequential weak completeness in Lebesgue-Bochner spaces \ \mbox{$L^p(X)(1 \le p < \infty)$} \ and Orlicz-Bochner spaces \ $L^{\varphi}(X)$.