Does time dilation increase within overlapping gravitational fields? If a spaceship was to travel to the barycenter of a binary star (the stars having identical size and mass) and come to a stop there, would the people on board this spaceship find that time is moving slower there (relative to the passage of time experienced outside of the gravitational fields of this binary star) due to the two stars overlapping gravitational fields at this point in space?  

Does time dilation increase within overlapping gravitational fields?
Gravitational time dilation depends on the observers' relative positions in the gravitational potential.

For relatively weak gravitational fields the rate at which a clock ticks for observer A compared to the clock rate for observer B is given by
$$\frac{d \tau_{\rm A}}{d\tau_{\rm B}} \simeq \sqrt{1 + \frac{2\Delta \Phi_{\rm AB}}{c^2}} \sim 1 + \frac{\Delta \Phi_{\rm AB}}{c^2},$$
where $\Delta \Phi_{\rm AB}$ is the difference in gravitational potential between observers A and B.

Gravitational potential is a scalar quantity, so the basic answer to your question is yes - the effects of the two neutron stars would be summative. However, I would hesitate to use the approximation above when dealing with the strong gravity near neutron stars.