We study approximation properties of additive random fields Y _d(t),t∈[0,1] ^d, d∈N, which are sums of d uncorrelated zero-mean random processes with continuous covariance functions. The average case approximation complexity n ^{Y _d} (ε) is defined as the minimal number of evaluations of arbitrary linear functionals needed to approximate Y _d, with relative 2-average error not exceeding a given threshold ε∈(0,1). We investigate the growth of n ^{Y _d} (ε) for arbitrary fixed ε∈(0,1) and d→∞. The results are applied to the sums of the Wiener processes with different variance parameters.