Test if two measurements come from normal distributions with the same mean?

I have two measurements $x_1\pm s_{x_1}$ and $x_2 \pm s_{x_2}$. The error intervals ($\pm$ one standard deviation) do not overlap. Does this necessarily mean that these are two samples from normal distributions with different means?

Or should I run a test to exclude (to some degree) the fact that they might still be from the same distribution?

Both means have an equal amount of measurements (about 2000). It is given that both are pulled from normal distributions.

• This reply does not appear to account for the supposition that $x_1$ and $x_2$ are accompanied with statements (or at least estimates--it's unclear which) about their errors. – whuber Nov 14 '11 at 20:41