I'm assuming by "degenerate states" you mean uncontrollable states. This essentially means that you cannot excite these modes (vectors is state space) from the input. An example of this might be a collection of masses connected in some complicated fashion with springs and we only have the ability to apply a force to one of the masses. You can imagine not being able to generate motion in every mass if we are limited to applying a force to only one mass. However, this is very different from a state being physically realizable. The difference is that we can give a system initial conditions to excite any mode we want. Think of the initial condition as a vector in state space. If we want to excite a particular mode (another vector in the state space) the initial condition vector must have a component in the direction of that mode.