Passive metamaterials have yet to be translated into applications on a large scale due in large part to their limited bandwidth. To overcome this limitation many authors have suggested coupling metamaterials to non-Foster circuits. However, up to now, the number of convincing demonstrations based on non-Foster metamaterials has been very limited. This paper intends to clarify why progress has been so slow, i.e., the fundamental difficulty in making a truly broadband and efficient non-Foster metamaterial. To this end, we consider two families of metamaterials, namely Artificial Magnetic Media and Artificial Magnetic Conductors. In both cases, it turns out that bandwidth enhancement requires negative inductance with almost zero resistance. To estimate bandwidth enhancement with actual non-Foster circuits, we consider two classes of such circuits, namely Linvill and gyrator. The issue of stability being critical, both metamaterial families are studied with equivalent circuits that include advanced models of these non-Foster circuits. Conclusions are different for Artificial Magnetic Media coupled to Linvill circuits and Artificial Magnetic Conductors coupled to gyrator circuits. In the first case, requirements for bandwidth enhancement and stability are very hard to meet simultaneously whereas, in the second case, an adjustment of the transistor gain does significantly increase bandwidth.