Submitted application online in February. 6 weeks later I had a phone interview. 2 weeks after that I had another phone interview (with a second researcher). Both phone interviews went well.
About three weeks later I received a job offer from a different institution. I e-mailed my RAND point-of-contact and told them I'd love to be able to consider RAND when making the decision. They called me back and said they would "rush" the scheduling to bring me out for an interview. I interviewed about a week later (3 months after I originally submitted the online application).
The interview was all day, from 8 am to ~4:30 pm. It involved about 7 one-on-one discussions with RAND researchers, a discussion with two researchers over a nice lunch, and my seminar on my postdoctoral research. My seminar was given in a conference room with ~6 attendees from the local office (Santa Monica), and about 3 or so attendees from other offices (Pittsburgh, Washington D.C.) via telecon and sharing my powerpoint screen over the web. 1.5 hours was allotted for my seminar. The attendees asked some good questions, but I was well prepared since the topic was my research of the past 3 years. (This type of all-day interview with a seminar presentation is typical in my field.)
The day ended with a discussion with HR about benefits and salary expectations. About 2 weeks later I received a phone call and a verbal offer. I asked for 2 days to consider, then called them back and accepted the offer. This was 3.5 months after my initial online application.
I'm not sure how long the process would have taken if I had not called them to notify them that I had another offer.