Time complexity of addition
If your algorithm uses asymptotically less than $n$ time, then it does not have enough time to read all the digits of the numbers it is adding. You are to imagine you are handling very large numbers (stored for example in 8MB text files). Of course, addition can be done very quickly compared to the value of the numbers; it runs in $\mathcal{O}(\log(N))$ time, if $N$ is the value of the sum.

This does not mean you can speed things up a little; if your processor handles 32 bits each operation, then you use $\frac{n}{32}$ time, but that is still $\mathcal{O}(n)$ and not $o(n)$.