The analysis of nanoindentation force curves collected on polymers through the common Oliver and Pharr procedure does not lead to a correct evaluation of Young's modulus. In particular, the estimated elastic modulus is several times larger than the correct one, thus compromising the possibility of a nanomechanical characterization of polymers. Pile-up or viscoelasticity is usually blamed for this failure, and a deep analysis of their influences is attempted in this work. Piling-up can be minimized by indenting on a true nanometer scale, i.e., at penetration depth smaller than 200 nm. On the other side, it is common knowledge that fast indentations minimize the effect of viscoelasticity. However, changing the indentation time in a broad range of contact time (fractions of second up to hundreds of seconds) did not allow the correct estimation of Young's modulus for the polymers used in this work. The final result is that the Oliver and Pharr procedure as well as any other procedure analyzing the unloading curve with elastic contact mechanics models cannot be employed to measure Young's modulus of polymers because its application is incorrect from a theoretical point of view, unless the analysis is limited to the very first nanometers of penetration depth when the contact is perfectly elastic. Viscoelastic contact mechanics models should instead be employed to characterize these materials.