Since the RIAA began the crackdown on individual users in September, some 400 warnings have been sent out.
The RIAA said 1,054 former file sharers have submitted affidavits for the group's amnesty program.
The new strategy of going after individuals -- which has drawn fire from some free-speech activists and others -- began after the industry failed to win legal battles to shut down so-called "peer-to-peer" networks that allow individuals to swap songs online.
The crackdown on unauthorized file-swapping comes amid a proliferation of legal sites offering music downloads for around one dollar per song.
"These lawsuits help to foster an environment that provides a level playing field for the growing number of legitimate online music services to thrive," the RIAA said in a statement.
The RIAA said the campaign has "made significant strides in deepening the understanding of the illegality of file-sharing copyrighted songs."
It cited a survey by Peter D. Hart Research Associates showing 64 percent of Americans understand it is illegal to download music, up from 37 percent in November 2002.