There are some small companies who have had their websites totally crash after a burst of unexpected media coverage so that no one could access their website until they upgraded to another tier of service or a different webhost .
As a matter of fact, I saw a documentary about this regarding the “Oprah Effect” where in the first few years, small entrepreneur were totally overwhelmed by a brief mention on her show. Subsequently, especially prior to her “Favorite Things” show each holiday season, producers from the show work with these small business owners months in advance to prevent a catastrophe.
Obviously, the first order of business is making certain that they have the web service structure in place to handle the enormous traffic load.
But for most small businesses and websites, the increase in traffic happens more gradually and the owners have time to adjust their hosting plans.