We provide a comprehensive impossibility result toward achieving string stability, i.e., keeping local relative errors in check with local controllers independently of the size of a chain of subsystems. We significantly extend the existing results, from the linear time-invariant (LTI) setting to any homogeneous controllers that can be nonlinear, time varying, and locally communicating. We prove this impossibility for a set of definitions with various norm choices, including the L-2-type standard and a bounded-input implies bounded-output (BIBO) type criterion. All the results hold for a general discrete-time controller, which should cover most applications.