We show that a multilayer analysis of the infrared c-axis response of RBa2Cu3O7-d (R=Y, Gd, Eu) provides important new information about the anomalous normal state properties of underdoped cuprate high temperature superconductors. Besides competing correlations which give rise to a pseudogap that depletes the low-energy electronic states below T*>>Tc, it enables us to identify the onset of a precursor superconducting state below Tons>Tc. We map out the doping phase diagram of Tons which reaches a maximum of ~180 K at strong underdoping and present magnetic field dependent data which confirm our conclusions.