It depends on what type of website you have. A static html site can probably get 10,000 daily visitors or more on a shared hosting server.
But one with a database such as web forum or blog might need to move to a vps virtual private server or dedicated server when it hits 2,000 daily visitors.
It also depends on the plugins you have on your forum or blog. Some plugins on forums and blogs can bog down a shared server.
If your website has a database such as a forum or blog. I'd be looking at moving the website to a vps or dedicated server once you hit 2,000 or more daily visitors.
At least get a vps for it. You don't want your website to go down or be slow as that can affect your websites earnings as well as seo rankings.
Google hates websites with lots of downtime and slow loading times.
Make sure your website is online and the website loads fast.