We study stochastic team problems with static information structure where we assume controllers have linear information and quadratic cost but allow the noise to be from a non-Gaussian class. When the noise is Gaussian, it is well known that these problems admit linear optimal controllers. We show that for such linear-quadratic static teams with any log-concave noise, if the length of the noise or data vector becomes large compared to the size of the team and their observations, then linear strategies approach optimality for "most" problems. The quality of the approximation improves as length of the noise vector grows and the class of problems for which the approximation is asymptotically not exact approaches a set of measure zero. We show that if the optimal strategies for problems with log-concave noise converge pointwise, they do so to the (linear) optimal strategy for the problem with Gaussian noise. And we derive an asymptotically tight error bound on the difference between the optimal cost for the non-Gaussian problem and the best cost obtained under linear strategies.