Global existence theorems for large data to hyperbolic fluid models very often have restrictions on the size of the relaxation parameter depending on the size of the initial data, and therefore leave the general question of well-posedness for large data open. In this note we show blow-up results to some corresponding model problems that underline the complexity of this question and hence also show the strength of global existence theorems for small parameters. (C) 2016 Elsevier Ltd. All rights reserved.