I think the original statement of Hartshorne was that for smooth $ X \subset \mathbb{P}^N$ with $ 3/2 \dim(X) > N$, then $X$ is a complete intersection (you recover your statement from this conjecture with the famous construction of Serre). Being a complete intersection implies that $ H^0(\mathbb{P}^N, O_{\mathbb{P}^N}(k)) \rightarrow H^0(X,O_X(k))$ is surjective for all $k$. A variety with such a surjective map for a fixed $k$ is called $k$-normal.

With surprisng new "topologico-geometric" methods, Fyodor Zak proved in the beginning of the 80's that if $3/2\dim(X) + 1 > N$ then $X$ is $1$-normal. He proved morever that for $ 3/2\dim(X) + 1 = N$ there exist only $4$ varieties which are not $1$-normal namely : the 2nd Veronese embedding of $\mathbb{P}^2$ projected down into $\mathbb{P}^4$, the Segre embedding of $\mathbb{P}^2 \times \mathbb{P}^2$ projected down into $\mathbb{P}^7$, the Plucker embedding of the grassmannian of $\mathbb{C}^2 \subset \mathbb{C}^6$ projected down into $ \mathbb{P}^{13}$ and the closed orbit of $E_6$ in $\mathbb{P}^{26}$ projected down into $\mathbb{P}^{25}$. You can read his wonderful book "**Tangents and Secants of Algebraic Varieties**" to get complete proofs of these facts.

If you restrict to the case of varieties defined by quadratic equations (which is a bit limitated but still interesting as a step toward a more complete answer), then Hartshorne conjecure has been proved in that setting by Ionescu and Russo (see http://arxiv.org/abs/0909.2763). Note also that Hartshorne conjecture completely fails for other toric projective Fano varieties, see http://arxiv.org/abs/1005.5546).