I have a dwarf with full plate armor and a tower shield with dex of 16 (+3). Full plate grants a max dex bonus of +1 to AC and the tower shield +2. Now, can I get both the +1 AND +2 to get a +3 dex bonus, or do I only get to take one and only have a dex bonus of +2 AC?
You take the lowest of the two maximum dex bonuses.
As GMJoe noted, they aren't giving you a dex bonus, they're setting a maximum on your dexterity bonus to AC. As in, you cannot have a higher dexterity bonus to AC while using that item. If you have armor with a max dex of +1 and a shield with a max dex of +2, then the highest dexterity modifier your AC can benefit from is +1. Even if your dexterity bonus is +9, your armor gets in the way and limits the benefit to your AC to +1.
From the SRD:
This number is the maximum Dexterity bonus to AC that this type of armor allows. Heavier armors limit mobility, reducing the wearer’s ability to dodge blows.
The armor and shield bonuses to AC do stack, however; if your armor gives a +9 armor bonus to AC and your shield gives a +2 bonus to AC, you would get a combined +11 to AC from them, in addition to other bonuses or penalties (the bonus from dexterity being capped at +1, as previously mentioned).