Do you have a message service on the POTS line?
1) I have a message service on my dial-up line - it indicates there is a voice message by interrupting the dial tone. Whereupon my FAX modem will refuse to connect because it is not detecting a continuous dial tone.
2) Silly question I know - is the FAX modem plugged into the "line" socket of the filter - won't work too well if plugged into the ADSL port (have actually seen this done because customer was using a "Y" adapter and it wasn't obvious where the modem was actually plugged in).
3) Have you had any thunder and lightning recently? Quite amazing how lightning can strike a modem and leave it looking like it's working (flashing front panel lights but no connectivity). It (the lightning) had actually burnt out a PCB track withing the isolated part of the modem.
When all else fails - have you tried swapping out the modem? I also need FAX very occasionally and purchased a USB FAX modem (powered from the USB port - about the size of a matchbox). Cost me about five quid and well worth the investment.
Good Luck!