Neutron stars and black holes
Observed neutron stars range from $1.0 \pm 0.1 M_{\odot}$ to $2.7 \pm 0.2 M_{\odot}$ according to table 1 of The Nuclear Equation of State and Neutron Star Masses, which lists dozens of examples. Keep in mind that the mass of the neutron star is typically substantially smaller than the mass of its progenitor star; late in the stellar life cycle a lot of mass is blown away, for instance a star that goes though an AGB phase may lose >50% of its mass. So our $1M_\odot$ Sun is likely to end up as a stellar remnant with $M < 1M_\odot$, probably a white dwarf.
According to Structure of Quark Stars, the mass is the only parameter to consider for neutron stars (but not hypothetical quark stars), although I would think rotation rate would be a factor.
This reference also states that neutron stars can be as small as $0.1 M_{\odot}$, but this does not imply that the sun will actually become a neutron star.
According to Possible ambiguities in the equation of state for neutron stars, it is the theory (equation of state) of neutron stars that is causing the current uncertainty about the limits of neutron stars.
Also, it is unknown whether or not neutron stars may become quark stars before becoming black holes. There is a term "quark nova" for such a hypothetical event.
Yes, there are absolute limits (with some theoretical uncertainty) for the mass of a progenitor star that can become a neutron star or black hole and the Sun is well below that limit.
The other answers here talk about the range of masses of neutron stars, but do not directly answer the question you pose: the answer arises from considerations of what happens in the core of a star during the course of its evolution.
In a star of similar mass to the Sun, core hydrogen burning produces a helium ash. After about 10 billion years, the core is extinguished and hydrogen burning in a shell results in the production of a red giant. The red giant branch is terminated with the onset of core helium burning, leaving a core ash of carbon and oxygen via the triple alpha process. After the core is extinguished again, there is a complicated cycle of hydrogen and helium burning in shells around the core. During this phase, the star swells enormously to become an asymptotic red giant branch star (AGB). AGB stars are unstable to thermal pulsations and lose a large fraction of their envelopes via a massive wind. The Sun is expected to lose about $0.4-0.5M_{\odot}$ at this time.
Now we get to the crux of the answer. What is left behind is a core of carbon and oxygen, with maybe a thin layer of hydrogen/helium on top. With no nuclear reactions going on, this core contracts as far as it is able and cools. In a star governed by "normal" gas pressure, this process would continue until the centre was hot enough to ignite carbon and oxygen burning (a higher temperature is needed to overcome the greater Coulomb repulsion between more proton-rich nuclei). However, the cores of progenitor stars with masses $<8M_{\odot}$ are so dense that electron degeneracy pressure takes over. The electrons in the gas are compressed so much that the Pauli Exclusion Principle results in all the low energy states being filled completely, leaving many electrons with very high energies and momenta. It is this momentum that provides the pressure that supports the star. Crucially, this pressure is independent of temperature. This means that the core can continue to cool without contracting any further. As a result it does not get any hotter in the centre and fusion never restarts. The final fate of stars like the Sun, and anything with a main sequence mass of $<8M_{\odot}$ is to be a cooling white dwarf. The figure of $8M_{\odot}$ is uncertain by about $\pm 1M_{\odot}$, because the details of mass loss during the AGB phase are not completely solved theoretically and it is difficult to empirically estimate the progenitor masses of white dwarfs.
Stars more massive than this have cores which do contract sufficiently to begin further stages of fusion, resulting in the production of an iron/nickel core. Fusion cannot produce any more energy from these nuclei, which are at the peak of the binding energy per nucleon curve, and thus the star will ultimately collapse and has a core mass greater than can be supported by electron degeneracy pressure. It is this collapsing core which forms a neutron star or black hole.
An interesting caveat to my answer is that there may be an evolutionary route for a star like the Sun to become a neutron star if it were in a binary system. Accretion from a companion might increase the mass of the white dwarf star, pushing it above the Chandrasekhar mass - the maximum mass that can be supported by electron degeneracy pressure. Though in principle this might form a neutron star, it is considered that a more likely scenario is that the entire star will detonate as a Type Ia Supernova, leaving nothing behind.
There are two questions here, namely about the limits on neutron star masses, and about the possibility of our sun becoming one. I'll try to argue that they are different questions, viz. the first about the stability and the second about the formation of such objects.
1) DavePhD's reference in the comments (here, for completeness) answers it completely. There is a lot of room for neutron star masses, because it depends intrinsically on the equation of state of nuclear (and possibly sub-nuclear) matter. Since we don't know the correct equation of state is hard to give strict boundaries. Without an equation of state one could have a mass as large as desired, just by increasing radius. So qualitatively the best one can do depends on the interplay between mass and radius, or density if you will.
The strictest limit comes from Schwarzschild radius, that is if you make too dense a star it would generate an event horizon and collapse into a black hole. Next to this, one notes that the speed of sound escalates with density, so if you try to make too dense a star it will have speed of sound greater than the speed of light, violating causality. This gives a limitation in the different equations of state possible. The upper bounds of about 3.5 solar masses comes from this consideration. You'll find all this more deeply discussed in the aforementioned paper. The summary is in Figure 3, page 51. I am completely ignorant of an analogous argument for lower bounds on the masses that use only some physical principles (in spite of my first, incorrect, answer that related it to angular momentum and Rob Jeffries kindly corrected me on the comments) so I have deleted the incorrect previous part.
2)Somewhat independently of the previous discussion, we can be pretty sure that the sun will never become a neutron star, no matter what equation of state is correct. This is because gravitational collapse of a star is a highly non-linear process, that besides the different nuclear fusion cycles, will generate shock waves. Therefore it will not proceed adiabatically, on the contrary this processes will shed most of a star's mass. Therefore to produce a neutron star we need to start with a very heavy one, typically of the order of tens of solar masses. This is the reason we attribute neutron star formation to supernova events.