The electromagnetic spectrum is a continuum of wavelengths of light, and we have labels for some ranges of these and numerical measurements for many.
Question: Is the EM spectrum continuous such that between two given wavelengths (e.g. 200nm and 201nm) there is an infinite number of distincts wavelengths of light? Or is there some cut-off of precision with which light might exist (e.g. can light only have wavelengths of whole number when measured in nanometers, etc.)?
In general the frequency spectrum for Electromagnetic (e.g light, radio, etc) is continuous and thus between any two frequencies there are an uncountable infinity of possible frequencies (just as there are an uncountable number of numbers between 1 and 2).
There are situations in which the only relevant frequencies are discrete (such as the modes in a cavity).
For any given experimental measurement you will always have a finite precision or bandwidth with which you can measure, and so although light at 200nm and 200.01nm is in principle different, you might not be able to tell in practice.
Formally there are an infinite number of different wavelenghts. However, any given physical system can only be found in a finite number of distinct physical states. To create a light source with a wavelength $\lambda$ that is well defined up to some resolution $\delta\lambda$, requires observing it within a system of size of the order of $\lambda^2/\delta\lambda$. So, the smaller we make $\delta\lambda$, the larger the system must be before we get physically distinct states within each such smaller interval.
Sir Elderberry, Punk_Physicist and the Count Iblis have all given correct answers in principle.
There are two phenomena (really thought experiment, rather than practical, devices) that one needs to heed.
A finite measuring time $T$ can only resolve frequencies to within an uncertainty of the order of $1/T$. This is the reciprocal relationship between the time spread $\Delta t$ in a pulse and the frequency spread $\Delta \omega$ in its Fourier transform, given by $|\Delta t|\,|\Delta\,\omega|\geq\frac{1}{2}$ as I show in my answer to the Physics SE question "Heisenberg Relation". This is the mathematical phenomenon underlying the Heisenberg uncertainty principle (but not the same as the latter: the latter arises because the Fourier transform relates a quantum state's expression in co-ordinates related by the Canonical Commutation Relationship). In practice, though, a finite pulse is modelled by a spread of frequencies over an interval, and all frequencies in the interval are present in the Fourier transform.
If you model a finite universe as a cavity, you will indeed get only a finite number of modes per frequency. So one aspect of the quantum light field ground state energy divergence is not really a divergence at all as the energy per unit volume is finite, as I discuss here. If you were Odin, say, creating a universe, it would cost you roughly a fixed energy per unit volume to make one like ours.
I believe that currently, light in free space (i.e., not in a waveguide or a crystal or anything tricky) is believed to be able to have all values of frequency/wavelength/energy. As you say, it is continuous. There are highly speculative theories that perhaps this is not true "all the way down" but thus far we have no evidence that the wavelength spectrum is discrete to my knowledge.
