  We consider solutions $u$ to the 3d nonlinear Schr\"odinger equation
$i\partial_t u + \Delta u + |u|^2u=0$. In particular, we are interested in
finding criteria on the initial data $u_0$ that predict the asymptotic behavior
of $u(t)$, e.g., whether $u(t)$ blows-up in finite time, exists globally in
time but behaves like a linear solution for large times (scatters), or exists
globally in time but does not scatter. This question has been resolved (at
least for $H^1$ data) if $M[u]E[u]\leq M[Q]E[Q]$, where $M[u]$ and $E[u]$
denote the mass and energy of $u$, and $Q$ denotes the ground state solution to
$-Q+\Delta Q +|Q|^2Q=0$. Here, we prove a new sufficient condition for blow-up
using an interpolation type inequality and the virial identity that is
applicable to certain initial data satisfying $M[u]E[u]>M[Q]E[Q]$. Our
condition is similar to one obtained by Lushnikov (1995) but our method allows
for an adaptation to radial, infinite-variance initial data that can be stated
conceptually: for real initial data, if a certain fraction of the mass is
contained in the unit ball, then blow-up occurs. We also show analytically (if
one takes the numerically computed value of $\|Q\|_{\dot H^{1/2}}$) that there
exist Gaussian initial data $u_0$ with negative quadratic phase such that
$\|u_0\|_{\dot H^{1/2}} < \|Q\|_{\dot H^{1/2}}$ but the solution $u(t)$
blows-up. We conclude with several numerically computed examples.
