Magnetic bilayers with different magnetic anisotropy directions are interesting for spintronic appli- cations as they offer the possibility to engineer tilted remnant magnetization states. We investigate the ferromagnetic resonance (FMR) linewidth of modes associated with two interlayer exchange- coupled ferromagnetic layers, the first a CoNi multilayer with a perpendicular magnetic anisotropy, and the second a CoFeB layer with an easy-plane anisotropy. For antiferromagnetic interlayer ex- change coupling, elevated FMR linewidths are observed below a characteristic field. This is in contrast to what is found in uncoupled, ferromagnetically coupled and single ferromagnetic layers in which the FMR linewidth increases monotonically with field. We show that the characteristic field at which there is a dramatic increase in FMR linewidth can be understood using a macrospin model with Heisenberg-type exchange coupling between the layers.