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.