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How are you supposed to handle a model having two (or more) invulnerable saves? Do you just take one or the other? Do they stack? Does the lower one just get -1 (so that a 5+ becomes a 4+)?

You take the better of the saves. For example, a terminator armor marine has a 5++, but if you give him a storm shield, he gets the 3++ from the shield.

Modifiers to invulnerable saves exist, but they are very clear that they modify an existing save.

Invun saves don't stack. Nor do cover saves, which are similar.
The model takes the best save, whether armour, invul or cover, once modified.

Should be in YMDC, but yea. One save.