In this paper, we study saddle-node bifurcation and its robustness analysis for a class of nonlinear systems with dynamic uncertainties. First, we formulate a robust bifurcation analysis problem of evaluating the region that contains all potential bifurcation points on which an equilibrium appears, disappears, or loses hyperbolicity depending on uncertainties. Next, we propose an existence condition of multiple equilibria and evaluation of their location. Then, we derive a condition for robust hyperbolicity of a set of potential equilibrium points, and identify the region that contains all potential bifurcation points. The proposed analysis method is applied to robustness analysis of a genetic network model representing a mechanism for generating induced pluripotent stem cells (iPS cells). We find that saddle-node bifurcation occurs in the iPS model. Then, by the proposed robustness analysis, we further show that the bifurcation is so robust that it plays an essential role for inducing pluripotency in actual iPS cells.