Some interesting data points I've gathered from myself and a few friends:

Some applications stay in the "Received" state (often for 5+ weeks) until they are processed. For these people, it seems like once your application moves to the "In Process" state, that's a signal that it will typically be processed, shipped, and delivered within a matter of days. Other applications move into the "In Process" state as soon as you apply (that same day or maybe the next day), and then stay in this state (again, often for 5+ weeks) until they are processed.

(My guess for why Dan got his passport so quickly is that there used to be some bugs -- likely since fixed -- in the new online renewal process where applications that somehow moved into "In Process" immediately were actually processed immediately, rather than experiencing the same typical waiting period.)

I'm not sure why there's a difference in how these applications get handled, but it seems preferable to have your application in the "Received" state. This is because, should your travel plans change and you suddenly need your passport sooner than anticipated, you can use the website to upgrade your application from routine to expedited processing while it's stuck in the "Received" state, but not while it's in the "In Process" state. (To do so, log in to MyTravelGov, click "Renew now," and then click on your existing application number. There should be an option to "Upgrade Processing or Delivery Options.") You will be charged an additional fee to upgrade to expedited processing. I don't know anyone who's tried to upgrade after-the-fact, so I'm not sure how quickly your renewal would be processed at that point.