The Schwarzschild radius is a way for us to calculate the extent to which something has to be shrunk in order for it to become a black hole but I was wondering is there a limit to the object that we shrink? Would it be possible for me to shrink a neutrino to some size that makes it into a black hole?
Due to the uncertainty principle, it's expected to be impossible to form a black hole smaller than about a Planck mass. Such a black hole would have a Schwarzschild radius approximately equal to the Planck length.
The idea is that if you were to confine a particle's position to be within a Planck length, its momentum uncertainty would be larger than the Planck momentum, and hence the mass-energy it contributes to your system would be larger than the Planck mass. That is, in order to confine your system to within a Planck length, you already have to put in at least a Planck mass worth of energy, so you just end up with a Planck mass black hole anyway.
This doesn't necessarily forbid a black hole from dropping below the Planck mass (and hence Planck length) through evaporation, however. That's a matter for a quantum gravity theory. Semiclassical calculations indicate that the lifetime of such a black hole would be only around 16000 Planck times, due to evaporation, but that figure could receive quantum gravity corrections.
(Side note: the Large Hadron Collider produces collisions with energies 15 orders of magnitude below the Planck scale, so it should not make any black holes. Ideas that it might are based on highly speculative models with "large extra dimensions". Under these models, consequences of string theory could manifest at energy scales far below the Planck scale. I'm not an expert on that, but see this physics.SE question and its hilariously negative answer.)
In classical General relativity, there is no limit.
However it is unclear exactly how a black hole with a mass less than one Planck mass would behave. (One Planck mass is about 0.02 mg, or roughly the mass of a mote of dust). Such small black holes would have a radius that is smaller than the Planck length, and would be quantum mechanical objects. However there is no quantum mechanical theory of gravity.
Very very small black holes are highly unstable, evaporating by Hawking radiation in a vanishingly short time.
Moreover, fundamental particles like neutrinos seem to be point-like. They have zero diameters (so they can't be shrunk). These are modelled well by quantum mechanics.