As Robin Chapman points out, this conjecture is probably false. Nonetheless, similar results have been obtained, usually using sieve theory. Montgomery and Vaughan have proven $$\pi(x+y)\leq\pi(x)+\frac{2y}{\log y}.$$

Combined with a standard Chebyshev estimate, this gives $$\pi(x+y)\leq\pi(x)+16\pi(y),$$ say (for all $x,y\geq2$). Erdos conjectured that $$\pi(x+y)\leq\pi(x)+(1+o_{y\to\infty}(1))\frac{y}{\log y}$$ which, combined with the prime number theorem, would give $$\pi(x+y)\leq\pi(x)+(1+o_{y\to\infty}(1))\pi(y).$$

This may be as close as possible, and some (such as Selberg) believe even this to be false, and the constant 2 in the first result mentioned is the best possible.

As Robin Chapman points out, this conjecture is probably false. Nonetheless, similar results have been obtained, usually using sieve theory. Montgomery and Vaughan have proven $$\pi(x+y)\leq\pi(x)+\frac{2y}{\log y}.$$

Combined with a standard Chebyshev estimate, this gives $$\pi(x+y)\leq\pi(x)+16\pi(y),$$ say (for all $x,y\geq2$). Erdos conjectured that $$\pi(x+y)\leq\pi(x)+(1+o_{y\to\infty}(1))\frac{y}{\log y}$$ which, combined with the prime number theorem, would give $$\pi(x+y)\leq\pi(x)+(1+o_{y\to\infty}(1))\pi(y).$$

This may be as close as possible, and some (possibly including Selberg, see comments) believe even this to be false, and the constant 2 in the first result mentioned is the best possible.