I think they are saying they couldn't connect as all lines were busy, so they would call someone they assumed was connected, causing the other person to drop off and free up a line, and then try and get that free line.
I think call waiting "beep beep" would also cause disconnects a lot of the time, if the person had that.
It was specifically the call-waiting beeps that interrupted the modem connection. I recall my 300 baud modem tolerating it, but 12/2400 (and higher) could not recover from the interruption.
You could set how long the modems would wait for reconnect. We had a service that offered free phone calls with 15 seconds interruption for ads now and then. Worked fine for free bbs time if you knew how to set it :)