Yes, this is true.  In 1952, [Nagura](https://projecteuclid.org/euclid.pja/1195570997) proved that for $n \geq 25$, there is always a prime between $n$ and $\frac{6}{5} n$.  Thus, let $p_k$ be a prime at least $25$.  Then $p_k+p_{k+1} > 2p_k$.  But by Nagura's result we have that $p_{k+2} \leq \frac{36}{25} p_k < 2p_k$.  Finally, one can easily check by hand that the result holds for small primes.