Even after years of study on random growth models, such as first- and last-passage percolation and directed polymers, much remains mysterious or out of reach technically. In particular, beyond the fundamental shape theorems guaranteeing linear growth rates for the passage times/free energy, there are sublinear fluctuations whose asymptotics are not established. Concerning the planar versions of these models, KPZ universality predicts order n^{1/3} fluctuations. In this talk, I will discuss new methods for obtaining a lower bound of \sqrt{\log n}, which (in a frustrating state of affairs) is the best known result in general. This is joint work with Sourav Chatterjee.