A renamed wither's bossbar/health bar does not change into a custom name. I renamed a wither into "Dinnerbone" and the whole wither flipped upside down, it has the name tag on it's head but the bossbar/health bar still has the name "Wither" even though the mob was renamed.
*To Reproduce
1. Rename wither
2. Point your crosshair at the wither and look at the bossbar

Observed Results
The custom name not showing on the wither's boss bar

Expected Results
The custome name should be showing on the wither's bossbar

Note: If you are playing versions 1.10, the bossbar can change if a wither is renamed but it doesn't do that in 1.11. You will notice that the name tag is shown on top of the wither once you point your cross hair at it but the boss bar is still defaulting to "Wither"

Causes:
It appears that in some betas starting from 1.11.0.5, the boss bar (withers, ender dragons and raids) has been lowered down due to the debug text (top of screen or possibly raid fixes) blocking the bar, it also removes the feature of custom names showing on the boss bar of the wither since they're the only boss to be renamed via name tags.