As Fiammetta pointed out, the historical Tiberius wasn't an admirable namesake. I'd imagine a Tiberius to be sharp, cunning, intelligent, and aggressive. Maybe a sports coach? It's funny to imagine a Tiberius in an Adidas tracksuit. It'd work well on a pitbull too.
masculine list:
https://www.behindthename.com/pnl/191050/124079feminine list:
https://www.behindthename.com/pnl/191050/124080