N-uniquely additive sets
Jelly, 7 bytes

œcS€ṢIP


Try it online!

Returns a positive number for truthy and zero for falsey.

œc       find combinations
  S€     sum each combination
    Ṣ    sort the sums
     I   find the difference between each pair of sums 
           iff any sums are the same, this returns a list containing 0
      P  product of the elements of the resulting list