(Sorry, I misread the question at first.) The following result reduces your question to a problem of analytic number theory:

**Theorem** (Hasse-Deuring-Waterhouse): For a prime $p$ and $N \geq 1$ the following are equivalent:

(i) There is an elliptic curve $E_{/\mathbb{F}_p}$ such that $\# E(\mathbb{F}_p) = N$.

(ii) We have $|N-(p+1)| \leq 2\sqrt{p}$.

As long as $p > 3$ (i.e., $p = p_n$ for $n \geq 3$) every elliptic curve can be put in "short Weierstrass form" $y^2 = x^3 + ax +b$.

So you are reduced to asking: is it true that for all $n > 3$ we have
$|p_{n+1} - (p_n+1)| \leq 2 \sqrt{p_n}$?

According to this esteemed source, having this kind of upper bound on the prime gap (always) is conjectured to be true but far from being proven. In fact, if I have it right this precisely Andrica's Conjecture. As Dror Speiser points out, it is not even known conditionally on the Riemann Hypothesis.

no(it follows from Serre's theorems on the image of the $\ell$-adic Galois representation.) $\endgroup$ – Pete L. Clark Dec 23 '15 at 23:44