I can't sign in to the New York Times website (mobile.nytimes.com) in Chrome or Dolphin (without Jetpack) when ADP is enabled. (It works in Firefox.)
I go to the login page, sign in, and I get back to the home page but the site still says "Subscribe | Log In" at the top. As soon as I disable ADP it works and the site recognizes I'm logged in.
The NYTimes app works fine, as does sign in via Firefox. Dolphin with Jetpack and its internal ADP plugin enabled (and the ADP Android app disabled) works fine.
I'm running Cyanogenmod 11S on a rooted OnePlus One phone, though this also happened on latest CM 11 M12 on a rooted Verizon Galaxy S4.