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 $\mathcal A^0 = \mathcal A$.

$\begingroup$ Indeed, both have the same free rank. Thanks for the correction. I'm aware of Mazur's result (which I think he proves in the appendix of the Eisenstein ideal paper) as well as yours (the link to the second paper appears broken), but haven't gone over all the details of the proofs. Is it reasonable to expect that such isomorphisms might also hold for more general squarefree conductors? Putting it differently, for abelian subvarieties $A$ of $J_0(N)$ where $N$ is squarefree, can we expect the torsion subgroup of $A(\mathbb{Q})$ to be contained within the cuspidal subgroup of $J_0(N)$? $\endgroup$ – Saikat Biswas Apr 23 '11 at 19:03

1$\begingroup$ Dear Saikat, Given the geographical location given in your profile, you probably know Amod Agashe's work. But just in case, the discussion preceding the statement of Thm. 1.1 in this paper of his suggests that the answer to your question about torsion subgroups is yes. Regards, Matthew  P.S. I fixed the broken link. $\endgroup$ – Emerton Apr 24 '11 at 1:30

$\begingroup$ Thank you so very much for reminding me about Amod's paper. In fact, as I now recall, he did briefly discuss the contents of that paper with me about two years ago as a prospective research topic. However, I never got around to reading the paper and started working on visibility instead. $\endgroup$ – Saikat Biswas Apr 24 '11 at 4:54
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.

$\begingroup$ Thanks for the paper and also for referring to the relevant section within it. $\endgroup$ – Saikat Biswas Apr 23 '11 at 19:09