  We develop a new general method for computing the decomposition type of the
normal bundle to a projective rational curve. This method is then used to
detect and explain an example of a Hilbert scheme that parametrizes all the
rational curves in $\mathbb{P}^s$ with a given decomposition type of the normal
bundle and that has exactly two irreducible components. This gives a negative
answer to the very old question whether such Hilbert schemes are always
irreducible. We also characterize smooth non-degenerate rational curves
contained in rational normal scroll surfaces in terms of the splitting type of
their restricted tangent bundles, compute their normal bundles and show how to
construct these curves as suitable projections of a rational normal curve.
