This symptom is caused by modems which have decided they don't want to report the true state of carrier, but instead want to always say that carrier is high. Apparently some systems won't even let you send commands to the modem unless carrier high, but for TSX this setting is not only stupid, it's wrong.
You see, TSX waits on a phone line until carrier goes high. Then it tries to log on a job. To be more specific, a process is created and begins taking commands from the command file associated with the line, such as BBS.CMD or START.CMD.
Normally this command procedure asks for a username, which of course the modem is not going to give, so TSX gives up after a while, logs the job off, drops DTR (expecting the modem to hang up the phone), then raises DTR back up and waits for carrier to go back up. WHOOPS! It's up already! Time to log on another job.... the great circle of life continues.
As best as I can recall, the setting needed it AT&C1, but there is no modem manual where I sit so I can't be sure. Somebody e-mail me and I'll update the page.