Can a prime number be bigger than the sum of adding the previous twin primes (other than 13)?

Let's modify the OP's observation as follows:

Let $(p,p+2)$ and $(q,q+2)$ be consecutive pairs of twin primes, e.g., $(107,109)$ and $(137,139)$. Then (conjecturally) $q\lt2p+2$.

This is, essentially, a Bertrand's Postulate for twin primes, and it's not hard to confirm that it holds for entries at the outset for the sequence $3,5,11,17,29,41,59,\ldots$. The basic explanation can be found in the heuristic twin-prime "theorem" $\pi_2(x)\sim2C_2x/(\ln x)^2$, although arguing that it (conjecturally) holds for all twin primes, not just for ones that are sufficiently large -- i.e., giving a (heuristic) twin-prime analog of the proof of Bertrand's Postulate -- seems problematic.

Remark: Amusingly, the modification proposed above of the OP's observation is technically agnostic with regard to the twin prime conjecture. Indeed, it would be easiest, in principle, to prove (or disprove) if there were an identifiable last pair of twin primes.