Something that I know, is that in any dimension N and for any |ψ⟩, the
pseudo-pure state
\begin{equation}
\epsilon|ψ⟩⟨ψ|+ \frac{1- \epsilon}{N}I_N
\end{equation}
is separable whenever
\begin{equation}
\epsilon > \frac{2}{n^2}, \epsilon > 0
\end{equation} (http://arxiv.org/abs/quant--ph/9811018).
Something similar can be said for Werner states (but in other direction). In principle, the Werner states are states which satisfy the reduction criterion but violate the Peres one. The Werner states $W_N$ for $N \times N$ system can be written as
\begin{equation}
W_N = (N^3-N)^{-1} ((N-\alpha )I + (N \alpha -1)V),
\end{equation}
where $V$ is the swap operator $V \psi_1 \otimes \psi_2 = \psi_2 \otimes \psi_1$. The states are inseparable for $\alpha < 0$ (sorry for the strange notation, it takes simpler form for a $2 \times 2$ system).
Regarding the Werner states and bound entanglement, it is conjectured (http://arxiv.org/abs/quant-ph/9910026) that there exist bound entangled states with non-positive partial transpose (so called, NPT states). Existence of such states would in particular imply
nonadditivity of distillable entanglement and would rule
out a simple mathematical description of the set of distillable
states (distillability is equivalent to so called n-copy distillability
for some n - formally, a state $\rho$ is n-copy distillable if n copies of $\rho$ can be locally projected to obtain a two-qubit NPT state).
The problem still remains open (for a long time it was regarded as a one of the most important problem in QI, now it is somehow forgotten) but since
that paper many partial results have been obtained. In particular it was shown
that it is enough to concentrate on the class of the Werner states as if there exist
NPT bound entangled states then there exist NPT bound entangled Werner
states (http://arxiv.org/abs/quant-ph/9708015v3).
To verify the conjecture you can probably concentrate on a particular
$4 \otimes 4$ Werner state which is the most entangled of the so-called suspicious Werner states. This state is conjectured
to be undistillable, so you can consider the condition for its n-undistillability and translate it to a condition called the half-property (see, http://arxiv.org/abs/0711.2613, for details)...
