Pixellated Bump Maps in 2.8

My procedurally generated bump maps appear to be quite low-res in eevee - is there a setting somewhere that I can improve this?

That’s a limitation of bump map for Eevee. Try using normal map. Check this:
https://docs.blender.org/manual/en/dev/render/eevee/limitations.html

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Not mentioned in the docs, there is a reason for having pixelation such as this.

High quality bump-map calculation as in Cycles is rather expensive to do using shaders and Eevee needs to remain interactive in the viewport or quick in F12 rendering, the current quality being used meanwhile is rather cheap to do.

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So the only way around it in eevee with procedurally generated textures is to render at a higher resolution then, yes?

That is what the docs. say, though rendering larger sizes will always use up more memory due to the image buffer.

See for more info also:

And that’s why I designed these shaders for EEVEE:

Just convert your bump map to a normal map. I use Krita for this task but there are different other tools available.

Yes pixaflux for example, but if you get tired to convert 260 bumpmaps into seamless normal maps, the EEVEE shaders mights be interesting.