Pilate was one of Sejanus' men. Sejanus was the head of the Praetorian Guard under Tiberius Caesar and became Tiberius' right hand man. As Sejanus became more powerful he got very ambitious and had designs to take out Tiberius and become Caesar himself. As part of these plans he began placing people loyal to him in positions of power throughout the empire. Pilate was appointed consul of Judea by Sejanus. Tiberius learned of Sejanus' plans and had him executed. Thus, anyone in the Roman empire who had any connection with Sejanus became very nervous.
Wow, I got an education in ancient history today!