By introducing carbon nanotubes (CNTs) into a lyotropic nematic liquid crystal, strongly enhanced viscoelastic behaviour results, allowing the extraction of very thin and long filaments in which the CNTs are uniformly aligned. The filament formation requires the liquid crystallinity of the host phase and it does not take place for coarsely dispersed nanotubes or if their concentration is below a threshold value. The type of nanotube plays little role, single-as well as multiwall CNTs both triggering the filament formation, but spherical C 60 fullerenes do not give rise to the phenomenon. We argue that individualized CNTs stiffen the rod-shaped micelles of the liquid crystal host and that elongational flow then increases the nematic long-range order as well as the micelle length. If the CNTs are present at a sufficient concentration to connect into continuous linear chains of arbitrary extension, the micelle stiffening is ensured regardless of length, taking the system into a positive feedback loop between increasing orientational order and diverging micelle length. It is this percolation-like transition to aligned and quasi-infinite micelles stabilized by chains of nanotubes that makes the filament formation possible.