Just to reinforce previous advice --
Making your page static HTML once it has been dug is the single best way to keep your site online throughout the traffic spike. Bandwidth is never the issue with these things -- it's Apaches choking on hundreds of concurrent connections or scripts spiking server loads to the point that it needs to be powercycled to recover.
Depending on the software you're using, and how popular the link becomes, you can potentially bring down the Apache service your site runs on (which will just kill your site and a small handful of others) or (worst case scenario) the whole webserver. We do have people watching server status constantly, but this will mean some unspecified amount of downtime, and possibly the site being temporarily disabled and a letter sent asking you to make the site static or install a plugin like WP-cache.
I've personally run into a guy whose page was killing an entire server after being dugg (this is on a quad Xeon with 4 gigs of memory, mind you) that was entirely fine as soon as I made the page static.
Don't rely on us to do this for you though, as most of the time we can't, particularly with softwares that make heavy use of .htaccess rewrite rules. Catching your diggs/slashdots quickly by analyzing comment/log spikes will help you keep up-to-speed, too.