10 = 5+3+2, all unique.
11 = 7+2+2 = 5+3+3, but neither sum has all three primes unique.
Extension: The problem above actually has been investigated thoroughly. Try also finding the largest integer in each range which can be written as exactly one such sum.
It was quickly pointed out to me that this question is a rewording of
Goldbach's Conjecture. The answer in all cases is 17. For the extension,
the answer in all cases is 40.
Other sites to investigate: