Yes, in an ideal world their system would only ever store and forward a received DMCA notice.
But here in the real world Dropbox doesn't really give a shit whether they actually got a real DMCA notice from Sony regarding the presence of "Spiderman.3.DVDRip.avi" in a particular 13-year-old's public folder — since none of those accounts are paid, they're just saving themselves money, and noone would discover it normally or have a reason to be pissed off at them. They can't even get in trouble for perjury since they were never sending DMCA notices to users, but rather telling users that Dropbox had received a notice.
But here in the real world Dropbox doesn't really give a shit whether they actually got a real DMCA notice from Sony regarding the presence of "Spiderman.3.DVDRip.avi" in a particular 13-year-old's public folder — since none of those accounts are paid, they're just saving themselves money, and noone would discover it normally or have a reason to be pissed off at them. They can't even get in trouble for perjury since they were never sending DMCA notices to users, but rather telling users that Dropbox had received a notice.