You can be smart, intelligent, and educated without being wise. In fact, it’s the norm.
Yes, familiarity with a field should improve skill, but I like to think that one can be smart, intelligent or wise in a situation that does not involve previous information about it.
Someone exposed to a brand new game or puzzle can be smart, intelligent or wise in solving it separately (to some degree) from having prior information.
By three methods we may learn wisdom: First, by reflection, which is noblest; Second, by imitation, which is easiest; and third by experience, which is the bitterest.