Recovery guarantee analyses of the tail-$\ell _{2,1}$ minimization approach applied to multiple measurement vector (MMV) model are presented. Exact joint sparse recovery from the MMV model benefits from the rank enrichment of $rank(\boldsymbol{X})$. Generally speaking, unique sparse solution exists for sparsity level $k < K_{0} \equiv [rank(\boldsymbol{X})+spark(\boldsymbol{A})-1]/2$. We observe in extensive empirical tests that the MMV tail-$\ell _{2,1}$ minimization approach is capable of recovering signals of the sparsity level up to $spark(\boldsymbol{A})$, even for $rank(\boldsymbol{X})\ll k$, significantly beyond the rank-enriched upper bound $K_{0}$. This phenomenon is now known to be the measure theoretical uniqueness solution of the MMV problem. To analyze the greater recovery capacity of the MMV tail-$\ell _{2,1}$ minimization approach, two necessary and sufficient conditions for the unique solution are established. It is shown that these two tail-$\ell _{2,1}$ solution conditions are more likely to hold than comparable conditions of the conventional $\ell _{2,1}$ minimization approach. Furthermore, theoretical recovery guarantee analyses of the tail-minimization approach are carried out. Specifically, two recovery probability estimates are derived. These successful recovery probabilities are shown to approach 1 not only as the number of measurements $L$ increases (exponentially) but also as $|T^{c} \cap S|\to 0$, where $T$ is an estimation of the support index set $S$. That the MMV tail-$\ell _{2,1}$ minimization algorithm can recover signals of sparsity level beyond $K_{0}$ and approaching $spark(\boldsymbol{A})$ is sufficiently illustrated from these recovery probabilities.