PACS: 14.60.Pq, 13.35.Hb, 14.80.Mz.-- LaTeX file using elsart style, 10 pages and 2 figures included.-- Final version.We study the interplay of neutrino oscillation and invisible decay in atmospheric and long-baseline neutrinos experiments. We perform a global analysis of the full atmospheric data from Super-Kamiokande together with long-baseline K2K and MINOS in these scenarios. We find that the admixture of v_μ -> v_τ oscillations with parameters Δm_32^2 = 2.6e-3 eV^2 and Θ_23 ~ 34º plus decay of the heavy neutrino, v_3, with lifetime of the order τ_3/m_3 ~ 2.6e-12 s/eV provides a reasonable fit to atmospheric neutrinos, although this solution becomes more disfavored (dropping to the 99% CL) once long-baseline data are included. Other than this local minimum, the analysis shows no evidence in favor of a non-vanishing neutrino decay width and an lower bound on the decay lifetime τ_3/m_3 > 9.3e-11 s/eV is set at 99% CL. In the framework of Majoron models, this constraint can be translated into a bound on the Majoron coupling to v_3 and an unmixed very light sterile state, |g_s3| < 8.6e-3 (2.2 eV/m_3).This work is supported by National Science Foundation grant PHY-0354776 and by Spanish Grants FPA-2004-00996, FPA2006-01105 and FPA-2007-66665-C02-01. MM is supported by MCI through the Ramón y Cajal program, by the Comunidad Autónoma de Madrid through the HEPHACOS project P-ESP-00346, and by the European Union through the ENTApP network of the ILIAS project RII3-CT-2004-506222.Peer reviewed