How long a website can stay on a shared hosting server depends on the website, the hosting account specs and the server itself.
If you're running a static html website then you could possible get 15,000 or more daily unique website visitors to that website and the shared hosting account might be able to handle it without a hiccup.
But if the website you're running has a mysql database such as a forum, or wordpress blog then the amount of visitors to your website on a shared hosting account will decrease dramatically.
With a wordpress website or forum that has a mysql database you'll most likely need to move your website over to at least a vps server once you're getting 2,000 visitors per day.
But if your hosting your website on an oversold shared server then you might start getting errors and downtime once you get to 1,000 visitors per day.
A non oversold shared server should be able to handle at least 2,000 daily unique visitors without any problems.
Once you get to 2,000 daily unique visitors coming to your website then you should start looking at moving your website over to a vps server.