The usual classical behaviour of S = 3/2, B-site ordered double perovskites generally results in simple, commensurate magnetic ground states. In contrast, heat capacity and neutron powder diffraction measurements for the S = 3/2 systems La2NaBO6 (B = Ru, Os) reveal an incommensurate magnetic ground state for La2NaRuO6 and a drastically suppressed ordered moment for La2NaOsO6. This behaviour is attributed to the large monoclinic structural distortions of these double perovskites. The distortions have the effect of weakening the nearest neighbour superexchange interactions, presumably to an energy scale that is comparable to the next nearest neighbour superexchange. The exotic ground states in these materials can then arise from a competition between these two types of antiferromagnetic interactions, providing a novel mechanism for achieving frustration in the double perovskite family.