*Following marshall's comment below, I (sadly) had to completely re-write my original answer.*

A famous open conjecture of Paul Erdos, first stated about 80 years ago, is that if all subset sums of an integer set $S\subset[1,n]$ are pairwise distinct, then $|S|<\log_2n+O(1)$ as $n\to\infty$. (Here $\log_2$ denotes the base-$2$ logarithm.) In modern terms, a subset of an abelian group, all of whose subset sums are pairwise distinct, is called *dissociated*. Similarly to Erdos' original problem, one can ask how large can dissociated subsets of other "natural" sets in abelian groups be. Say, you are asking what is the largest possible size of a dissociated subset of the set $\{0,1\}^n\subset{\mathbb R}^n$, and this particular problem has been studied by a number of authors. It is known that the largest size of its dissociated subset is
  $$ \frac12(1+o(1))\,n\log_2 n; $$
see, for instance, [this paper by Bshouty][1] for details and a historical account.

[1]: http://www.cs.mcgill.ca/~colt2009/papers/004.pdf