We prove that in any unitary CFT, a twist gap in the spectrum of operator product expansion (OPE) of identical scalar quasiprimary operators (i.e. phi x phi) implies the existence of a family of quasiprimary operators O t,l with spins l ->.infinity and twists tau -> 2 Delta phi in the same OPE spectrum. A similar twist-accumulation result is proven for any two-dimensional Virasoro-invariant, modular-invariant, unitary CFT with a normalizable vacuum and central charge c > 1, where we show that a twist gap in the spectrum of Virasoro primaries implies the existence of a family of Virasoro primaries Oh, (h) over bar with h -> infinity and (h) over bar -> c-1/24 (the same is true with h and (h) over bar interchanged). We summarize the similarity of the two problems and propose a general formulation of the lightcone bootstrap.