Motivated by recent experimental observations [Parker, Hua, and Chin, Nat. Phys. 9, 769 (2013)1745-247310.1038/nphys2789], we analyze the stability of a Bose-Einstein condensate (BEC) in a one-dimensional lattice subjected to periodic shaking. In such a system there is no thermodynamic ground state, but there may be a long-lived steady state, described as an eigenstate of a "Floquet Hamiltonian." We calculate how scattering processes lead to a decay of the Floquet state. We map out the phase diagram of the system and find regions where the BEC is stable and regions where the BEC is unstable against atomic collisions. We show that Parker et al. performed their experiment in the stable region, which accounts for the long lifetime of the condensate (∼1s). We also estimate the scattering rate of the bosons in the region where the BEC is unstable. © 2014 American Physical Society.