The Linux distribution provided for BeagleBoard is not a hard real-time kernel as defined by Wikipedia. The interrupt response times are not deterministic.
However, "real-time" means different things to different people. When someone asks about the “real-time” capabilities of a system, it is important to understand the requirements they are trying to meet. They might need a system that can process video at 30 fps or process an audio stream without interruptions. The BeagleBoard can do both. For control applications, the BeagleBoard can probably handle a 1KHz loop, but anything below that is probably not possible.