All good nucleophiles are strong bases, but the opposite is not always true because nucleophilicity and basicity aren't always the same. As the steric hindrance around a base increases, its nucleophilicity decreases, but its basicity is not affected. So OH- is a strong base and good nucleophile, but OC(CH3)3- is a strong base but a bad nucleophile.
In the end, bases attack H atoms and nucleophiles attack carbon atoms. So you can see why OH- is a strong base: it can very easily attack H atoms. It is also a strong Nu because it can easily displace a good LG and attach onto a C atom. Looking at OC(CH3)3-, it is strong base and can take H atoms, but its too big (sterically hindered) to attach onto a C atom and therefore is not a good Nu. (a base like this would lead to elimination, not substitution)
You also have to look at the solvent you are using. The trends for better LG, better Nu and strong base are all for aprotic solvents. The trends reverse for protic solvents due to interactions of the base/Nu with the solvent.
I hope that makes sense. 🙂
ps G1SG2 responded while I was typing my response, so sorry if its redundant!