We study the pointwise rate of divergence of the sequence of discrepancy sums S_i(x) = ∑_j=0^i-1 (ϗ_[0,1/2)-ϗ_[1/2,1))(x+j θ) for (Lebesgue) generic rotation parameter θ. Almost-sure upper and lower bounds for the rate of divergence are given by a Khinchin-like criterion related to convergence of a certain integral. Concluding remarks address the impossibility of finding a generic asymptotic rate of divergence as well as the related study of how frequently S_i(x)=0.