Unless you have excessive play in the spindle, it would be nearly impossible to turn an eccentric. Even then, the symptom would be rough finish and chatter.
Play can easily be checked bu mounting a rigid shaft in the chuck and bringing a dial indicator into contact. If there is play, you will see the reading change as you push or pull on the chuck.
Three jaw chucks usually don't locate work concentric with the spindle axis, That is why we use four jaw chucks and dial in the work. Depending on the chuck and how much wear it has, you can see eccentricity of .010" or more. The lathe bearings should be preloaded so they will be slightly warm after use and there should be no axial play.