The 3LBBs do not explicitly state that bonuses never stack. However, it is implied that they don't stack, based on some explicit instructions for specific situations. For example, take this description of the Protection from Evil spell:
This spell hedges the conjurer round with a magic circle to keep out attacks from enchanted monsters. It also serves as an "armor" from various evil attacks, adding a + 1 to all saving throws and taking a - 1 from hit dice of evil opponents. (Note that this spell is not cumulative in effect with magic armor and rings, although it will continue to keep out enchanted monsters.)(Men & Magic, p. 23)
So, Protection from Evil definitely doesn't stack with magic armor or with magic rings. What about magic rings? Here's the description of the RIng of Protection:
A ring which serves as +1 armor would, giving this bonus to defensive capabilities and to saving throws.(Monsters & Treasure, p. 33)
This is a little more ambiguous, in two ways. It doesn't say that a magic ring doesn't stack with magic armor, but the name "Ring of Protection" suggests it ought to be an item that creates a circle of protection, like the spell. On the other hand, the phrase "serves as +1 armor" might mean it simply adds a defense bonus, the way magic armor would, or it might mean that it's effectively the same as wearing magic armor +1. I believe I've seen one edition define the Ring of Protection the latter way.
But let's look at magic armor and magic shields:
If the shield's bonus is greater than that of the armor there is a one-third chance that the blow will be caught by the shield, thus giving the additional subtraction.(Monsters & Treasure, p. 31)
So, magic shields don't stack, either. Part of the time, you find a set of armor and shield, with a single, uniform bonus. Occasionally, you find just armor or just a shield, and in rare circumstances, you find the Shield +3, the only way to get a +3 defensive bonus in the 3LBBs (there is no Armor +3 listed.)
Given the fact that shields don't stack with armor and protection spells don't stack with armor or rings, I interpret the rules as sayings rings don't stack with armor, either; you get the best magic defense bonus available, not the sum of all possible bonuses. Why do it that way? Because it's consistent and easy to remember. Either they all stack, or none of them stack. Do it another way, and you have to remember which stacks and which doesn't (and, in some rules systems, how much they stack.)
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