Pebble game lower bound?
The time hierarchy theorem gives, for every $k$, a function in P with a runtime lower bound of $\Omega(n^k)$. Unfortunately, this is not enough to separate P from NP: to do this, we need a function in NP with a superpolynomial runtime lower bound.

One popular approach for tackling the P vs NP question is via circuits. The best lower bounds for explicit functions are only linear. Perhaps this is the context in which you heard of superlinear lower bounds as a seeming barrier.