  Although $f(R)$ modified gravity models can be made to satisfy solar system
and cosmological constraints, it has been shown that they have the serious
drawback of the nonexistence of stars with strong gravitational fields. In this
paper, we discuss whether or not higher curvature corrections can remedy the
nonexistence consistently. The following problems are shown to arise as the
costs one must pay for the $f(R)$ models that allow for neutrons stars: (i) the
leading correction must be fine-tuned to have the typical energy scale $\mu
\lesssim 10^{-19}$ GeV, which essentially comes from the free fall time of a
relativistic star; (ii) the leading correction must be further fine-tuned so
that it is not given by the quadratic curvature term. The second problem is
caused because there appears an intermediate curvature scale and laboratory
experiments of gravity will be under the influence of higher curvature
corrections. Our analysis thus implies that it is a challenge to construct
viable $f(R)$ models without very careful and unnatural fine-tuning.
