To take a slightly different tack on this chucking problem, it seems extremely difficult to rechuck anything in a 4 jaw and to indicate it back to true running. I think the condition of the chuck jaws and slideways has more to do with this than actual bearing faults. As tightening pressure increases, the tiny bit of wear and slack in the jaw slideways, and possible worn taper effect on the jaw grips becomes significant. The increased tightening pressure may at some point not move the part predictably, instead causing a degree of tilt to occur which spoils the setup accuracy attained thus far.
A tilted circle is an ellipse and will not indicate round.
First of all, it is necessary to indicate the part in two spots, either a face and an annular ring, or two annular rings spaced some distance apart. You achieved much the same result by switching to a chuck and steady rest.
To use the 4 jaw alone will require the use of a hammer and an indicator. The only way to align the face of the part (if available) is with the hammer, to tilt the part. If two annular rings are used, then adjust the chuck to center the closest annular ring, and tap the part with the hammer to center the outermost ring.
It is necessary to keep going back and forth as an adjustment to either position affects the other one. But you can keep splitting the difference until you get down as close as you have the patience for. IF you find this is impossible to do, then your chuck is no good (assuming your part is straight) and needs to be reground. You may find that the 'natural position' of the chucked part is a few thou off 6" out, but that you can persuade it into position with a hammer. For precision work, this is still not the best condition to start out with. Rework your chuck.
Collet chucks may also suffer this misalignment phenomena a wee bit. 5C collets are bad for this, because the easy fit at the back end of the collet makes the outer end of a long piece float off center. You can tilt a part chucked in a collet with your hammer as well. If you cannot get zero runout a few inches from the collet, then suspect that your collet is no good. Get some real Hardinge collets.
Chucking a part between chuck and tailcenter is also prone to 'getting you' when doing precision work. The 4 jaw may have a mind of its own, and inexperienced users may think that plugging a tailcenter into the end of the part negates the effect of the part being misaligned with the axis of the lathe. Not so, the part will flex the tailcenter, and machining in this condition will create an offsize diameter with a bit of taper thrown in for bad effect.
Turning between chuck and tailcenter is best accomplished by chucking on a very narrow band on the end of the piece. Then, this acts as a crude universal joint, and the tailcenter may then have enough influence to actually align the part with the rotational axis of the lathe, and it will cut straight and repeatably.
First you get good, then you get fast. Then grouchiness sets in.
(Note: The opinions expressed in this post are my own and are not necessarily those of CNCzone and its management)