R(2)BaMO(5) (R = Gd or Y; M = Cu or Zn) oxides have been studied using the specific heat, DC magnetic Susceptibility and electron paramagnetic resonsance (EPR). For one member of the series without a magnetic moment at M, namely Gd2BaZnO5, measurements reveal long-range antiferromagnetic order at T-N (Gd2BaZnO5) = 2.3 +/- 0.1 K, much lower than the Curie-Weiss temperature Theta of 15.9 +/- 0.3 K. This indicates the existence of competing interactions that introduce a large degree of magnetic frustration in the system. For Y2BaCuO5 the Cu-Cu interactions are responsible for the broad maximum in the magnetic contribution to the specific heat centred at 18.5 +/- 0.01 K that stretches beyond the instrumental limit of 25 K. The strong Cu-Cu interactions also present in Gd2BaCuO5, combined with the Gd-Cu interaction, polarize the Gd sublattice, giving measurable contributions to the specific heat in the same temperature range. In addition, they broaden the Gd EPR line and saturate its integrated intensity. The ordering temperature of Gd ions is raised to T-N (Gd2BaCuO5) = 12.0 +/- 0.1 K.