TECH: Modem & phone line not speaking

[Mary Lee Freeman <mlf@TENET.EDU>]

Dear Tech People,
I think you may be my last hope.  I have 2 modems and 2 phone line exten-
sions--one in my office and one in the library.  All was fine until
the phone man decided that he needed the Library's ext. for the
elevator phone.  Many "no dial tone" messages later I found that
he had been in that same afternoon and had disconnected the Library
ext. for elevator use.  His solution was to hook up an ext. which I
share with the other library across campus on a "first-come" basis.
This is necessary because the phone system has been "maxed out" for
connections.  I'm sure you can tell that I don't really understand
that part!  The Library modem has never worked on the shared ext.
Instead of no dial tone messages I get "no carrier" messages.  It
whines just as it did before, but instead of beeps or tones after
that (which I was expecting) I hear a "twittering" noise such as I
have heard when dialing a fax machine with a regular telephone (by
mistake), followed by No Carrier.
What I have tried--Brought in a single-line phone to try in the other
modem connection slot to test the phone line.  It worked.
Bought two 25-ft. ext. phone cords, hooked them together, and was
able to call out using the Library modem on the office extension.
This also worked in reverse--could call out using the office modem on
the shared Library ext.  All the cords (25's and regular short ones)
worked when connecting the office modem to the office ext.  None
worked when connecting the Library modem to the Library ext.  Both
modems use script files for dialing and no settings have been changed
(at least not intentionally.  I did notice that this computer had a
check on Save Settings on Exit).  The Library modem is internal so
I don't know the brand.
What can you suggest?  It's too cumbersome to hook up 50 feet of
phone cord from the office to the Library modem each time someone
wants to dial out, since it runs through a doorway and behind the
circ. desk.
Any assistance is appreciated.
Mary Lee Freeman
mlf@tenet.edu
Carrollton, TX
PS--I've tried the Diagnostics in Comit Lite and it thinks things are
just fine.