We consider implications of dynamical Borel-Cantelli lemmas for rates of growth of Birkhoff sums of non-integrable observables \(\phi(x)=d(x,q)^{-k}\), \(k>0\), on ergodic dynamical systems \((T,X,\mu)\) where \(\mu(X)=1\). Some general results are given as well as some more concrete examples involving non-uniformly expanding maps, intermittent type maps as well as uniformly hyperbolic systems.