It can. It's easy to check in the warm waters of the Med - next time you're swimming, dive down to the bottom of the rudder and wobble it left to right. Don't be gentle, the waves aren't!
If it moves, you will feel it and hear it - crew on deck can compare the sound to what you hear when underway (for the person underwater all things sound very different). A tiny bit of play is okay and doesn't make sounds, but if it notably moves left and right (or forward/aft - try that too), then most likely one or both of your rudder bearings are worn out. Usually the bottom one.
You can find several threads on replacing them around here somewhere, most recently done by MagicalArmchair, I think.