Hi Jonathan,
Many thanks for helping me out with this!
After reading your hints I did 2 things:
– 1: Creating a new Blender test-file with a high resolution.
– 2: I used the test photo and edited it in Photoshop.
The UV map of my original model might be large (4k), but the UV’s of the texurepainted faces might be small in the bigger picture. Maybe that this created the lower resolution (pixelated bump map). To test this I started a new Blender file and used the edited photo. In this Blender file I made a plane and a map of 4k. In this case the resolution of the bump map was much better. Beside, the bump map (edited like you said) was much cleaner.
Thanks again!