After all problems we went thru with our setup we’ve solved it ourselves by buying a new CAT6 cable. Strange thing is that all computers do work with the same cable.
But for future reference, I’ll leave the complete story online:
We run our network from a Draytek Vigor 2910. We bought a printer from Saturn a month or two ago. But, as it turns out HP does not support that router. We bought a Photosmart C6280, a multifunctional printer from HP. The printer does not get an IP-address from the router, nor from one of the hubs connected to the router. All other devices ( laptops, pc’s, Macs, … ) don’t run into any problems, everything works.