The digg effect will usually not have much of an effect on larger websites but small websites may be shut down quickly due to the server not being able to cope with the high amount of requests or because all of their bandwidth is consumed.
it creates a massive swarm of traffic to the website that was "dugg", causing the servers that the website is hosted on to become extremely slow and laggy
Person One: dude, my website just got to the front page of digg!
Person Two: soon, you'll be suffering from the digg effect
Techie: You fool. You have fallen victim to the Digg effect. I'll let Dell know your computer is on its way.