I made the texture 2048x2048 in GIMP. The odd lines are caused by GIMP's blur. I'm not killer with nodes, so how do I get the height map not to show these odd lines?
These "lines" are the border lines between different values on the height map. The Bump node converts grayscale gradients to normal map angles and the gradients only work on these borders (or value steps). Since you are using a 8 bit image you only have 256 different values to play with. To make things worse, you are only using a small part of that gamut (you don't use fully white to fully black) so you end up with about 70 grayscale values.
So what you need to do is to take your height map back to what ever image tool you use, convert it to a 16 or 32 bit image, adjust the levels so that the information uses the full gamut and blur the edges away.
This problem is a handling problem, not anything that a specific tool introduces.
Turns out Eevee has limits. https://docs.blender.org/manual/en/dev/render/eevee/limitations.html#materials the info about 8 bit n 32 bit was educational .I'll use that info from now on.