How can a low-mass star increase its mass to 1.4 Msun? In my astronomy class I learned that a only low-mass stars (&lt; 0.5 Msun) will contract, and then become degenerate, until it is a white dwarf. However, we also learned about the Chandrasekhar limit, which states white dwarfs have a maximum mass of 1.4 Msun. 

If the starting star has less than 0.5 Msun, how does the resulting white-dwarf reach up to 1.4 Msun?

Am I perhaps wrong in my assumption that the mass of the resulting white-dwarf depends on the mass of the original star?
You were taught wrong. Stars of up to about 8 solar masses will end up as white dwarfs. But it is only their cores that become degenerate and end up as the white dwarf. The rest of the envelope is lost during the giant phase due to a dense wind.

There is a non-linear, but probably monotonic, relation between the initial progenitor and final white dwarf masses (see below, from Kalirai 2013) - the Sun will likely end as a 0.5 solar mass white dwarf, but in normal stellar evolution, degenerate white dwarf stars can only be produced up to about 1.25 solar masses by the most massive progenitors. Any more massive than this and it is likely that the core does not become degenerate before igniting and burning through the heavier nuclear fuels. The most massive, probably single, white dwarf known is "WD 33" in the cluster NGC 2099 and has a mass of $1.28^{+0.05}_{-0.08}\ M_{\odot}$, is likely made of an O/Ne mixture, and had an estimated progenitor mass on the main sequence of $&gt;3.5\ M_{\odot}$ (Cummings et al. (2016).

In order to get a more massive white dwarf, up to the Chandrasekhar mass (about 1.38 solar masses for a C/O or O/Ne white dwarf in general relativity), it almost certainly needs to accrete mass from a close binary companion or be the result of some sort of merger.

This is the leading candidate to explain type Ia supernova.