Is relation following given characteristic already in 2NF, 3NF and BCNF?
Consider a relation schema R(A, B, C), with the functional dependencies:

A → B
B → C


In this schema there is only one candidate key (A), and it has a unique attribute. But the schema is not in BCNF, since B is not a superkey, neither in 3NF, since C is not a prime attribute.

So your intuition is correct.