If both neutron stars and white dwarf stars can have the same mass, what determines what a star of that mass will become when it "dies"? My understanding is that roughly 1.4 solar masses is the upper limit for white dwarf stars, and that the lower bound for neutron stars is around 1.1 solar masses. Is there any way to tell what a star will form upon death, knowing that its mass will end up being in between these two bounds? 
 A: Yes, there are theoretical models of stellar evolution that tell us what to expect.
Broadly, we expect that stars with an initial mass less than 8 solar masses ($8M_{\odot}$) will end their lives as white dwarfs. So I think there is a misconception in your question - the progenitors of white dwarfs and neutron stars are usually a lot more massive than what ends up in the stellar remnant. So a star of initial mass $1.1<M/M_{\odot} <1.4$ will always end up as a white dwarf.
The reason for this $8 M_{\odot}$ upper limit is that below it, the cores of such stars never achieve the temperatures required for carbon fusion. Instead, electron degeneracy pressure is able to support the carbon/oxygen core (of mass about $\leq 1.1M_{\odot}$), whilst the outer envelope is lost in a stellar wind and planetary nebula. (Note that white dwarfs more massive than this need to have accreted mass, usually as part of a binary system).
Stars with initial mass larger than $10M_{\odot}$ do not form an electron degenerate core and are able to contract and heat up sufficiently to ignite carbon and subsequent elements until a core of iron peak elements is formed. This may then collapse to form a neutron star or possibly a black hole for very massive stars.
There is a grey area at $8-10M_{\odot}$, where it may be possible to form massive oxygen/neon white dwarfs, or they might explode as electron capture supernovae leaving behind neutron stars - it just depends how massive the core can become and whether the oxygen is able to ignite in a degenerate configuration. The remnants here, whether they be white dwarfs or neutron stars could have very similar masses.
Either way, although these models are well understood, there are sufficient theoretical uncertainties (at the $\pm 1 M_{\odot}$ level), that observational tests and empirical confirmation of the exact relationship between initial mass and the type and mass of the remnant is still desirable.
