You can probably make a system that detects backlog at the funnel or belt leading into the chest, but I don't think there is a way to use the inspector panel to detect if the chest is full. In the next update, there will be a way to configure the inspector panel, so then you will be able to set its threshold to the capacity of the chest and detect fullness in that way. For now, fullness has to be detected indirectly.
You could probably have a logic gate detect if there are items on the input belt and start some sort of timer contraption, then if the items haven't been cleared out by the end of the timer, it means that the chest is full.