Let $A$ be an abelian variety over a number field $K$ and consider the Neron model $\mathcal{A}$ of $A$ over $X=Spec{\mathcal{O}_K}$. If $\mathcal{A}^0$ is the identity component of $\mathcal{A}$, then $\mathcal{A}^0$ is an open subgroup scheme of $\mathcal{A}$ that fits into a short exact sequence $$0 \rightarrow \mathcal{A}^0 \rightarrow \mathcal{A} \rightarrow \Phi_A \rightarrow 0$$ over $X$. Considering these smooth group schemes as sheaves for the flat (fpqf) topology over $X$, the associated long exact sequence of flat cohomology groups begins $$0\rightarrow\mathcal{A}^0(X)\rightarrow\mathcal{A}(X)\cong{A(K)}\rightarrow\Phi_A(X)\rightarrow\ldots$$ where the indicated isomorphism follows from the Neron mapping property. Now, we know that $A(K)$ is finitely generated i.e. it has a free part (copies of $\mathbb{Z}$) and a torsion part (which is finite). Since $A(K)/{\mathcal{A}^0(X)}$ is contained within $\Phi_A(X)$ (which is finite), it follows that the group $\mathcal{A}^0(X)$ of global sections of $\mathcal{A}^0$ must contain all of the free part of $A(K)$. My question is  is it possible in this general scenario to determine how much of the finite part of $A(K)$ is captured by $\mathcal{A}^0(X)$ or do we need additional information on $A$?
First, a remark: the free part of $A(K)$ is a quotient, not a sub, and so it is possible that a point of infinite order in $A(K)$ could have nontrivial image in $\Phi_A(X)$. Probably what you mean is that $\mathcal A^0(X)$ and $A(K)$ have the same free rank. Regarding torsion, my interpretation of your question is that you are asking about the map $A(K)_{tors} \to \Phi_A(X)$, and are curious is to whether or not it can have a kernel (so that some part of $A(K)_{tors}$ is contained in $\mathcal A^0(X)$). As far as I know, this varies a lot depending on the particular abelian variety, but in particular cases it has been quite intensively studied. For example, if $p$ is prime and $A =J_0(p)$ is the Jacobian of the modular curve $X_0(p)$, then Mazur showed in his Eisenstein ideal paper showed that the map $A(\mathbb Q)_{tors} \to \Phi_A( \mathrm{Spec}\ \mathbb Z)$ is an isomorphism. I generalized this to subabelian varities $A$ of $J_0(p)$ in my paper here. For an example in some sense opposite to this, see this paper of Conrad, Edixhoven, and Stein, in which they show that if $A = J_1(p)$ (again $p$ is prime), then $\Phi_A(\mathrm{Spec}\ \mathbb Z)$ is trivial, so that $\mathcal A^0 = \mathcal A$. 


I will have to wait for a more reasonable hour to give a complete answer, but I believe this paper of mine  joint with X. Xarles  is relevant to your question. Most of it works in the case of an abelian variety over a local field  in this case we get fairly definitive results and you can see what's happening. In the case of a global field, what one has is mostly the Szpiro Conjecture and its analogues. See $\S 6$ of the paper for a (brief, breezy) discussion of such conjectures for abelian varieties over number fields. 

