8 choose 2 is 28. So the obvious upper bound is 28 prime sums. But definitely can go lower because a prime number (ignoring 2) must be an odd number. So we can only get a prime number by adding an odd number and an even number. E.g.

Four odd, four even = 16 possible odd sums

Five odd, three even = 15 possible odd sums

Six odd, two even = 12

Seven odd, one even = 7

All odd = 0 possible odd sums

So a firmer upper bound is 16 possible prime sums.

Don't know if can go lower or not