Given a fixed vector bundle E on P^1, one can ask: what is the moduli space of rational curves in P^n with normal bundle E? For projective 3-space, well-known results of Ghione-Sacchiero and Eisenbud-Van de Ven prove that the space of curves with given normal bundle in P^3 is irreducible of the expected dimension, and Eisenbud and Van de Ven conjecture that the same thing holds for arbitrary P^n. Alzati and Re found a single counterexample to this conjecture in P^8. In this talk, I describe joint work with Izzet Coskun finding an infinite family of counterexamples to the conjecture, where we show that the moduli spaces of rational curves with fixed normal bundle can have arbitrarily many components.