In many geometrically generated random network topologies it is a common phenomenon that the expected degree of an average node tends to infinity with the network size, whenever asymptotic connectivity is required. This is clearly an obstacle to scalability, as a real node cannot handle an unbounded number of links within bounded processing time. We call it the lack of degree scalability. To investigate this phenomenon, we set up a general modeling framework that contains many different random graph models as special cases. In this framework we identify two conditions and prove that whenever they are present, they make the lack of degree scalability unavoidable. As our general conditions are directly checkable in most specific cases, even in complicated ones, they can serve as powerful tools to show that a possibly complex random network topology model lacks degree scalability. Often this would otherwise be rather hard to prove via direct analysis of the stochastic geometry of the model.