Try these two options first before adding new servers. They may allow you to stick with one server, but your results may vary.
For speeding the site up when you are hit with many concurrent users, look into installing the APC PECL extension (http://us2.php.net/manual/en/book.apc.php). APC will allow you to cache the compiled version of your scripts, saving the step of the PHP interpreter running each time a script is executed.
Also, if you are experiencing heavy load on the database server, look into installing memcached and caching database results for a certain time period, if possible (http://us2.php.net/manual/en/book.memcache.php).
Finally, if you do decide to get a separate server, look into possibly getting a dedicated SQL box. This, of course, assumes that your application is a database heavy application, as web apps are these days. Segregating SQL into a separate box allows it to take advantage of all of the resources on that box, with more cache and processing power. It could be the way to go.