The presence of crystallographic defects can induce notable effects on the mechanisms ruling the current transport in metal/semiconductor contacts. In this context, in this Letter, the impact of stacking faults (SFs) on the characteristics of 4H-SiC Schottky diodes was investigated under both forward and reverse bias. In particular, in the presence of SFs under the contact, while no significant effect on the ideality factor and barrier height was observed under forward bias, an anomalous increase in the leakage current occurred under reverse bias. The observed behavior of the leakage current could be explained by a space-charge limited current model, consistent with the presence of a distribution of trapping states in the gap of 4H-SiC. An increase in the reverse bias above 30 V leads to a complete trap filling. The weak temperature-dependence of the leakage current observed in this regime suggests the coexistence with a tunneling of the carriers through the barrier. The results can be useful to understand unexpected failures in 4H-SiC Schottky diodes.