We prove that every strong null sequence in a Banach space X lies inside the range of a vector measure of bounded variation if and only if the condition $mathcal{N}_1(X,{ell}_1)={Pi}_1(X,{ell}_1)$ holds. We also prove that for $1{leq}p<{infty}$ every strong ${ell}_p$ sequence in a Banach space X lies inside the range of an X-valued measure of bounded variation if and only if the identity operator of the dual Banach space $X^*$ is ($p^{prime}$,1)-summing, where $p^{prime}$ is the conjugate exponent of $p$. Finally we prove that a Banach space X has the property that any sequence lying in the range of an X-valued measure actually lies in the range of a vector measure of bounded variation if and only if the condition ${Pi}_1(X,{ell}_1)={Pi}_2(X,{ell}_1)$ holds.