So my understanding of electron degeneracy is that as the area an electron is decreased, its oscillations increase in speed. Is there a point that it is moving at just under light speed and cannot move faster?
Let's say we have a white dwarf. We add enough material to it, that it reaches the maximum degeneracy pressure. If we add more mass to it, what happens?
There are infinitely many higher kinetic energy states for the squeezed electrons to occupy. Their speeds will tend asymptotically to $c$, but in relativity, the kinetic energy is $(\gamma -1)mc^2$, where $\gamma = (1 -v^2/c^2)^{-1/2}$, and has no upper limit.
Therefore there is no maximum electron degeneracy pressure if we are considering an ideal gas of non-interacting fermions.
It is possible you have become confused by some explanations of the Chandrasekhar limit. For ideal degeneracy pressure, and Newtonian gravity, the limit occurs when an infinite central pressure is required to support the star, which requires the electrons to have infinite kinetic energy - i.e. they would all have speeds of $\sim c$.
In practice, the real Chandrasekhar limit occurs at a finite density and a slightly lower mass. This is either because (a) General Relativity includes the electron kinetic energy on the right hand side of the hydrostatic equilibrium equation, causing an instability at finite density; (b) the nuclei present in the white dwarf start capturing the electrons when they have high enough energies to cause inverse beta decay, which destabilises the star; (c) nuclear reactions take place between closely packed nuclei and set off a supernova; (d) some combination of the above - all of (a), (b) and (c) occur at quite similar densities of $\sim 4 \times 10^{13}$ kg/m$^3$ in a typical C/O white dwarf, which would have a mass of about $1.38M_\odot$.
To address your specific question - what happens if you add more mass to a white dwarf that has a central density just below one of the above thresholds? I think there are essentially two possibilities. One is that nuclear reactions are ignited, either in the core or further out in a helium envelope; the fusion burning spreads throughout the star rapidly and there is enough energy released to unbind the star and explode it as a type Ia supernova. A second possibility is that a collapse occurs ("accretion-induced collapse"), the increasing density and electron energy cause rapid inverse beta decay and the collapse will only be halted when the nuclei dissolve into a sea of nucleons (mostly neutrons) and a strong (force) repulsive interaction supports the object as a newborn neutron star.
