With the advent of RFM, the Accept/Regret process has been replaced by Preferential Ranking system on most campuses. The A/R process is done after Chapters have submitted invitation lists. Pref Ranking occurs after parties are completed and before invitations lists are turned in.
Regret With Interest is supposed to tell Panhellenic and the Chapters that a PNM would have accepted had she been allowed more choices. One of the problems I've seen is that some PNMs would Regret With Interest even though they couldn't stand XYZ. They thought it made them look better, less picky, more open minded, etc. The PNM can later say "I didn't reject lousy, crappy GPA ranking, non-competitive, low tier, party animals or study nerds,
fill-in-the-blank XYZ. I would have accepted if my party list wasn't already full."

On many campuses the possibility of later getting an invitation from somewhere you previously regretted is slim. But when it happened the PNM wasn't happy and she would simply regret them again, this time without interest.