The 2D layered organic-based magnet [Fe(TCNE)(NCMe)2][FeCl4] (TCNE=tetracyanoethylene) exhibits a unique macroscopic magnetic bistability between the field-cooled and zero-field-cooled states, which cannot be explained by either superparamagnetic behavior or spin freezing due to spin glass order. This magnetic bistability is described through consideration of the ensemble of uncoupled 2D Ising layers and their magnetization reversal initiated by a field-induced nucleation of magnetic bubbles in individual layers. The bubble nucleation rate strongly depends on the external field and temperature resulting in anomalous magnetic relaxation.