fft - What determines the accuracy of the phase result in a DFT bin?

What are the factors that affect the accuracy and precision for the phase that is given by the DFT?

Just thinking medium-hard about this, it occurs to me that it must have something to do with the window length $L$ and the sampling frequency $F_s$, and maybe the fundamental frequency that I'm interested in extracting?

I'm going to keep working on some of the math, but I'd like to start a discussion and get some insight for which way to look.