For magnets with a fully frustrated interlayer interaction, we argue that the quantum phase transitions from a paramagnetic to an antiferromagnetic ground state, driven by pressure or magnetic field, are asymptotically three dimensional, due to interaction-generated nonfrustrated interlayer couplings. However, the relevant crossover scale is tiny, such that two-dimensional behavior occurs in an experimentally relevant low-temperature regime. In the pressure-driven case the phase transition may split, in which case an Ising symmetry related to interlayer bond order is broken before magnetism occurs. We discuss the relation of our results to recent experiments on BaCu Si2 O6.