"Kerkrade is a city that is not in Drenthe but in Limburg."
Translation:Kerkrade is een stad die niet in Drenthe maar in Limburg ligt.
Is it wrong to say "...die in Drenthe niet ligt..."? Obviously I'm still learning and don't know for sure, but it sounds right to me.
It's wrong because it is a place. In your sentence the focus is on niet ligt, which leads to the question what is it doing then? Standing? Sitting?
Have a look at this on the implications of the position of niet in a sentence, it also gives examples: http://www.dutchgrammar.com/en/?n=WordOrder.45