A fundamental property of Physical Unclonable Functions (PUFs) is uniqueness, which results from the intrinsic characteristics of each PUF instance. However, PUF architectures employ elements whose physical characteristics and behavior may be very similar among different instances, thus leaking unwanted information. We explore the consequences of this effect by mounting Template Attacks over XOR Arbiter PUFs. In the attack, Challenge-Respose Pairs (CRPs) are profiled in one FPGA instance of the PUF to predict responses of a different FPGA instance, obtaining up to 80% of accuracy. We show that replicating the same attack strategy with a well-known Machine Learning (ML) algorithm would not be as effective, since different PUFs instances will not share similar CRP sets. Our template attack only needs few CRPs for profiling (at most 170), but it can be applied to different instances without additional training, which Machile Learning cannot do with unbiased PUF instances.