I consider N=2 pure super Yang-Mills of gauge group $SU(2)$ in four dimensions. It is one of the simplest four dimensional interacting quantum field theories whose part of the mass spectrum, the BPS spectrum, is known exactly at the quantum level (and everywhere in the moduli space of vacua: at weak coupling because the BPS condition implies that the semiclassical results are exact (Olive, Witten, 1978), and at strong coupling by Seiberg, Witten, 1994).

My question is:

What is known about the non-BPS part of the spectrum?

For example: is the non-BPS spectrum non-empty?

I understand that the BPS states are much more easier to study but I would like to have some global picture of what the non-BPS part looks like, in particular what are the relative "sizes" of the BPS and non-BPS spectra.

I have taken N=2 super Yang-Mills of group $SU(2)$ as a specific example but the question makes sense for any four dimensional N-supersymmetric QFT with $N \geq 2$ (example: what about N=4 on the Coulomb branch?) and also for other dimensions (example: N=(2,2) in two dimensions).