Mine does this also and in my case it's the rubber flaps that support the cup are not as pliable as they once were and are folded down in such a way that they impede the motion of the cupholder.
If I bend the rubber flaps upward it works as it should for a short while.
I've used this cupholder everyday for over 10 years and I'm amazed that I haven't broke it. Knocking on wood. I'm sure my knowing that it costs around $170 for a new one causes me to be extra careful with it.