The synthesis and crystal structures of a family of decametallic Cr(III) "molecular wheels" are reported, namely [Cr10(OR)20(O2CR')10] [R' = Me, R = Me (1), Et (2); R' = Et, R = Me (3), Et (4); R' = CMe3, R = Me (5), Et (6)]. Magnetic studies on 1-6 reveal a remarkable dependence of the magnetic behaviour on the nature of R. In each pair of complexes with a common carboxylate (R') the nearest neighbour CrCr magnetic exchange coupling is more antiferromagnetic for the ethoxide-bridged (R = Et) cluster than for the methoxide analogue. In complexes 2, 4 and 6 the overall coupling is weakly antiferromagnetic resulting in diamagnetic (S = 0) ground states for the cluster, whilst in 1 and 5 it is weakly ferromagnetic thus resulting in very high-spin ground states. This ground state has been probed directly in the perdeuterated version of 1 ([D]1) by inelastic neutron scattering experiments, and these support the S = 15 ground state expected for ferromagnetic coupling of ten Cr(III) ions, and they also indicate that a single J-value model is inadequate. The ground state of 5 is large but not well defined. The trends in J on changing R are further supported by density functional calculations on 1-6, which are in excellent agreement with experiment. The very large changes in the nature of the ground state between 1 and 2, and 5 and 6 are the result of relatively small changes in J that happen to cross J = 0, hence changing the sign of J.