In how many possible combinations can you add 5 numbers?

Your question basically boils down to computing subsets of the set $\{A,B,C,D,E\}$ that consist of at least 2 elements, since each such subset (for example the set $\{A,C,E\}$ corresponds to one way of forming a sum.

The number of all subsets of a set with $n$ elements is $2^n$ (for every element you have two options: include it in the subset or leave it out, which gives you $2^n$ different options to form a set). There is one subset with $0$ elements, the empty set, and there are $n$ subsets with $1$ element (the singleton sets), so that indeed gives us the answer $2^n-n-1$, or in your specific case of $n=5$, we get $32-5-1=26$.