Your site shouldn't immediately drop connections when the maximum number of concurrent connections has been reached, unless your host has something in place specifically to handle this case (see below). In Apache, the MaxClients directive is used to set the maximum number of concurrent connections. Anything beyond this limit will go into a queue to be processed after a pending request is finished, the size of which is determined by the ListenBacklog directive and the operating system.
To keep the queue moving along as fast as possible, is generally recommended to ensure that the KeepAlive directive is set to "On," allowing multiple HTTP requests in succession (for instance scripts, CSS, images, etc) to be served over the same TCP connection without the overhead of recycling it every time. Often you can check for KeepAlive being used in the response headers, though not all servers will respond with KeepAlive headers even after establishing a persistent connection. You could, however, ask if your host uses KeepAlive if you don't see these headers coming back.
All that being said, ab can be used to benchmark performance on higher-than-average load conditions, to see how your server will behave. If your current load results in unacceptably high page load times, it may be time to relocate. In most cases, you should notice this well before requests actually start to timeout.