As far as I am aware, this is the fastest 16 bit adder not to use pistons.
It takes 10 ticks (1 second), and is ideal if you find pistons too unreliable or noisy. It can add two numbers, each number being up to about 65000, so anything larger than 16 bit is not really necessary unless you plan to work with really massive numbers.

Additional Notes

BUGFIX: Unless the file you downloaded is called "adder 1_0", you have an old version with bugs.
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I made a mistake, it is still 10 ticks.
I do not think any more improvements with this design are possible (as far as speed is concerned at least), so this should be the last version. I think it is now bug free.
The video is pretty bad, but should give you some idea of what the adder is like.

Not pipelineable, but could be made to be by adding repeaters / changing repeater delays.
Red for inputs and blue for outputs.

If you find a problem, tell me.