In this paper, we study the local times of vector-valued Gaussian fields that are 'diagonally operator-selfsimilar' and whose increments are stationary. Denoting the local time of such a Gaussian field around the spatial origin and over the temporal unit hypercube by Z, we show that there exists lambda is an element of (0, 1) such that under some quite weak conditions, lim(n ->+infinity) (n)root E(Z(n))/n(lambda) and lim(x ->+infinity) -log P(Z>x)/x(1/lambda) both exist and are strictly positive (possibly +infinity). Moreover, we show that if the underlying Gaussian field is 'strongly locally nondeterministic', the above limits will be finite as well. These results are then applied to establish similar statements for the intersection local times of diagonally operator-self-similar Gaussian fields with stationary increments.