When wearing armor that reduces player health, more damage can be received than is visible. At three hearts, for example, the remaining seven hearts of the default ten can still be damaged despite not being visible. Additionally, the health bar will at randomly switch from the intended amount of health to the default ten hearts, both visibly and in functionality. Included is the world file and screenshots of the problem.
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That's why the issue is so annoying. You can't reliably lose or gain health without removing and replacing the piece of armor in question.
Some problems were fixed in the 1.6 prerelease. The armor now seems to display the proper number of hearts at all times. However, the player will still have "20 HP" upon wearing the armor for the first time. Damage in this state seems to fix this though.

Is this still a concern in the current Minecraft version? If so, please update the affected versions in order to best aid Mojang ensuring bugs are still valid in the latest releases/pre-releases.
In the 1.6 release, I can confirm that the bug has been fixed; the armor reliably lowered my health and did not reset to 20 hearts when I died or paused.

I'm going to close this. Maxwell, if it's not close able, you can reopen it.
Confirmed. However it happens inconsistently; walking into a cactus will cause the health bar to act like there's an extra (non-visible) heart. The Wither Effect acted consistently as if there were only three hearts until after a while, when the full health bar began to randomly reappear. The hearts remained visible until I took off and put the armor back on. Pausing and unpausing seems to be the biggest cause of making the hearts visible.