Can Anonymous really pull off a DDOS attack on the whole Internet?

In a word, yes.

Which is to say, it’s technically possible, and using software specifically designed to take advantage of a “reflection” flaw in DNS itself. Whether Anonymous actually commands the skill to pull this off is debatable, for the moment. It’s not likely to remain theoretical for much longer, however.

Read a discussion of their announcement at http://prohackingtricks.blogspot.com/2012/02/anonymous-launches-ddos-attack-on.html.

See the actual declaration by Anonymous at http://pastebin.com/NKbnh8q8.