Is there a way to generate a list of distinct numbers such that no two subsets ever have an equal sum?

For $n$ servers, consider weights $$ 1 + m, 2 + m, 4 + m, \ldots, 2^n + m $$ for $m$ large enough to make sure each is less than twice each of the others.

The uniqueness of subset sums for subsets of equal size follows from the uniqueness of binary expansions.

If the weights needn't be integer, you can choose them from the set $$\left\{1+\frac1p:p\text{ prime}\right\}$$