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 $3/2\dim(X) + 2 1 > N$ then $X$ is $1$-normal. He proved morever that for $3/2\dim(X) + 2 1 = N$ there exist only $4$ varieties which are not $1$-normal namely : the 2nd Veronese embedding of $\mathbb{P}^2$, \mathbb{P}^2$projected down into$\mathbb{P}^4$, the Segre embedding of$\mathbb{P}^2 \times \mathbb{P}^2$, mathbb{P}^2$ 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 $\mathbb{P}^{26}$. \mathbb{P}^{26}$projected down into$\mathbb{P}^{25}$. You can read his wonderful book "Tangents and Secants of Algebraic Varieties" to get complete proofs of these facts. If you restrict to the case of varieties defined by quadratic equations (which is a bit limitated but still interesting as a step toward a more complete answer), then Hartshorne conjecure has been proved in that setting by Ionescu and Russo (see http://arxiv.org/abs/0909.2763). Note also that Hartshorne conjecture completely fails for other toric projective Fano varieties, see http://arxiv.org/abs/1005.5546). 3 added 1 characters in body I think the original statement of Hartshorne was that for smooth$ 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$3/2\dim(X) + 2 > N$then$X$is$1$-normal. He proved morever that for$ 3/2\dim(X) + 2 = N$there exist only$4$varieties which are not$1$-normal namely : the 2nd Veronese embedding of$\mathbb{P}^2$, the Segre embedding of$\mathbb{P}^2 \times \mathbb{P}^2$, the Plucker embedding of the grassmannian of$\mathbb{C}^2 \subset \mathbb{C}^6$and the closed orbit of$E_6$in$\mathbb{P}^{26}$. You can read his wonderful book "Tangents and Secants of Algebraic Varieties" to get complete proofs of these facts. If you restric restrict to the case of varieties defiend 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). 2 deleted 4 characters in body I think the original statement of Hartshorne was that for smooth$ X \subset \mathbb{P}^N$with$ 3/2 \dim(X) + 2 > 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$3/2\dim(X) + 2 > N$then$X$is$1$-normal. He proved morever that for$ 3/2\dim(X) + 2 = N$there exist only$4$varieties which are not$1$-normal namely : the 2nd Veronese embedding of$\mathbb{P}^2$, the Segre embedding of$\mathbb{P}^2 \times \mathbb{P}^2$, the Plucker embedding of the grassmannian of$\mathbb{C}^2 \subset \mathbb{C}^6$and the closed orbit of$E_6$in$\mathbb{P}^{26}\$. You can read his wonderful book "Tangents and Secants of Algebraic Varieties" to get complete proofs of these facts.