No idea what Dragonfire Inspiration, but to answer your question in general terms it strictly depends on what kind of bonuses the performances grant. Two instances of Inspire Courage wouldn't as they give competence for hit/dam and morale for fear saves; morale and competence bonuses don't stack with other morale or competence bonuses. On the other hand Inspire Courage would stack with Archivist bard's Naturalist performance would as it gives insight bonuses, i.e. a different bonus type. These would have to come from multiple Bard's though as you can only have one performance up at a time without certain spells and in my example Archivist loses Inspire Courage for Naturalist.