I found out what the problem is:
If you have two of one kind of armor piece, sometimes the game will confuse the two. So, for example, if one shielded left T45b Arm is in perfect condition, and the second one is broken, the game might get them confused. The game thinks you want to equip the broken one instead of the one in good condition, but since you are trying to equip a armor piece from the menu that has some health left, the game doesn't know what to do.
In other words, keep only one piece of each kind of armor, fellow wastelanders. Remember, you can always paint em a different color if you have several models and want to carry them with you.