Answer: No.

Assume that there exist five such prime numbers. If there are three among them pairwise incongruent modulo $3$, then by adding each of the three separately to the sum of the other two we get three distinct sums modulo $3$. One of those three is $0$ modulo $3$ and therefore is not a prime number. However, if among the five we only get two distinct remainders when dividing by $3$, then by the pigeonhole principle there must be three among them that are congruent modulo $3$. The sum of those three, however, is divisible by $3$ and therefore not a prime number.