  We investigate static, spherically symmetric solutions in gravitational
theories which have limited curvature invariants, aiming to remove the
singularity in the Schwarzschild space-time. We find that if we only limit the
Gauss-Bonnet term and the Ricci scalar, then the singularity at the origin
persists. Moreover we find that the event horizon can develop a curvature
singularity. We also investigate a new class of theories in which all
components of the Riemann tensor are bounded. We find that the divergence of
the quadratic curvature invariants at the event horizon is avoidable in this
theory. However, other kinds of singularities due to the dynamics of additional
degrees of freedom cannot be removed, and the space-time remains singular.
