Synthetic polymer materials are rapidly replacing more traditional inorganic materials such as metals and natural polymeric materials such as wood. Since these novel materials are flammable, they require modifications to decrease their flammability through the addition of flame-retardant (FR) compounds. Recently, environmental regulation has restricted the use of some halogenated FR additives, initiating a search for alternative FR additives. Nanoparticle fillers are highly attractive for this purpose since they can simultaneously improve both the physical and flammability properties of the polymer nanocomposite. We show that carbon nanotubes can surpass nano-clays as effective FR additives if they form a jammed network structure within the polymer matrix, such that the material as a whole behaves rheologically like a gel. We find this kind of network formation for a variety of highly extended carbon-based nanoparticles: single and multi-walled nanotubes, as well as carbon nanofibers.