Two B-site ordered double perovskites, La2LiReO6 and Ba2YReO6, with S=1 were investigated as geometrically frustrated antiferromagnets, using x-ray and neutron diffraction, superconducting quantum interference device magnetometry, heat capacity, muon spin relaxation (μSR), and Y 89 magic-angle spinning (MAS) NMR. La2LiReO6 has a monoclinic structure (P21/n) with cell parameters at room temperature; a=5.58262(22) Å, b=5.67582(20) Å, c=7.88586(27) Å, and β=90.240(4)°. A zero-field cooled/field cooled (ZFC/FC) divergence at 50 K was observed in the susceptibility. The ZFC susceptibility is zero below ∼5 K for polycrystalline samples, suggesting a cooperative singlet ground state but weak moments are induced by cooling in very small fields ∼1 mT. No evidence of long-range ordering is evident in heat capacity, neutron-diffraction, or μSR data. The ZF spin dynamics from μSR are anomalous and can be fitted to a stretched exponential rather than the Kubo-Toyabe form expected for random frozen spins but the muon spins are decoupled in longitudinal fields (LF), consistent with spin freezing of the fraction of spins relaxing within the muon time scale. The internal fields sensed by the muons are anomalously small, consistent with an electronic spin-singlet state. Ba2YReO6 is found to be cubic (Fm3m) with cell parameter a=8.36278(2) Å at 300 K with no change in symmetry at 3.8 K, at variance with the Jahn-Teller theorem for a t2g2 configuration for Re5+. Y 89 MAS NMR shows a single peak indicating that Y/Re site disorder is at most 0.5%. The susceptibility shows two broad peaks around 50 and 25 K but no evidence for long-range order from heat capacity, neutron diffraction, or μSR. The ZF μSR result shows a two-component ground state with both slow and fast relaxations and decoupling results in a 1 kG LF, indicating spin freezing. These results are in sharp contrast to the long-range AF order found in the S=3/2 isostructural materials, La2LiRuO6 and Ba2YRuO6, indicating that the reduction to S=1 plays a major role in ground state determination.