Energy cannot decent during optimization despite non-zero gradient

Assume we have an (at least) 2nd-order differentiable energy $f(x), xin R^n.$ And $n$ is very big. Mathematically, I think it is impossible to find a point $bar{x}$ where the energy cannot be decreased along its negative gradient when the gradient is not equal to zero. However, such kind of case can be occasionally found when an energy is optimized numerically, such as using the interior point method. In the optimization software, there is always a return code that describes that "the step size is too small and cannot make progress". Is it because of the numerical issues? Or is it that my mathematic conclusion is wrong? If it is from numerical issues, what kind of issue could lead to this problem?