In systems biology, a common approach to model biological processes is to use large systems of nonlinear differential equations. The associated parameter estimation problem then requires a prior handling of the global identifiability question in a realistic experimental framework. The lack of a method able to solve this issue has indirectly encouraged the use of global sensitivity analysis to select the subset of parameters to estimate. Nevertheless, the links between these two global analyses are not yet fully explored. The present work reveals new bridges between sensitivity analyses and global non-identifiability, through the use of functions derived from the Sobol' high dimensional representation of the model output. We particularly specify limits of variance-based sensitivity tools to completely conclude on global non-identifiability of parameters in a given experimental context.