We prove central limit theorems, strong laws, large deviation results, and a weak convergence theorem for suitably normalized occupation times of critical binary branching Brownian motions started from Poisson random fields on $R^d, d \geq 2$. The results are strongly dimension dependent. The main result (Theorem 2) asserts that in two dimensions, as opposed to all other dimensions, the average occupation time of a bounded set with positive measure converges in distribution to a nondegenerate limit.