In this work, we investigated the impact of crystallographic defects (specifically stacking faults, SFs) on the mechanisms of the current transport in 4H-SiC Schottky contacts. The electrical characteristics were studied under both forward and reverse bias. In particular, while the presence of SFs under the contact did not show a significant impact on the forward characteristics of the Schottky diode, a significant increase in the leakage current occurred under reverse bias in defective diodes. This anomalous behavior can be explained by a space-charge limited current model, consistent with the presence of a trapping state distribution in the 4H-SiC gap. An increase of the reverse bias above 30 V leads to a complete trap filling. The weak temperature-dependence of the leakage current observed at highest voltage suggests that a tunneling of the carriers through the barrier can be also present.