My experience is that the blue line appears for what the website thinks is a stop instead of a connection, and it can be wrong. It will definitely appear between flights where you turn around, for example, between an LAX-HNL and HNL-LAX. However, when I had an LAX-HNL (5 hour stop) HNL-SFO, I didn't get a blue line. Also, the computer attempted to check me in for every segment until the next blue line.

By the way, I had a fare break between LAX-HNL and HNL-SFO, but still no blue line.