The condition number for the stiffness matrix of any method (finite elements, finite volumes, finite differences) applied to a second order differential operator always grows as ${\cal O}(h^{-2})$ where the exponent equals the order of the differential operator. As a consequence, if you just make the mesh fine enough, you can make the condition number arbitrarily large -- and in fact, $10^4$ is a pretty small value that probably results from the fact that you have a relatively small number of cells or mesh points.
The reason why you don't observe this issue is because for time dependent (dynamic) problems, the matrix that needs to be inverted is either of the form $M+\Delta t\, A$ (parabolic problems with one time derivative) or $M+\Delta t^2\, A$ (hyperbolic problems with two time derivatives). The mass matrix has condition number ${\cal O}(1)$ and so the total matrix you need to invert has condition number of either ${\cal O}(1+\Delta t\,h^{-2})$ or ${\cal O}(1+\Delta t^2\,h^{-2})$ in the two cases above. In other words, the multiplication of the bad matrix by a small number $\Delta t$ makes the problem better conditioned.
