I did it from various generations of the Nokia Communicator. The phone would initiate a special kind of phone call (from memory, the same AT string used to start an ISDN connection) and then send digital data over the link, not audio tones directly.
This would emerge onto the analogue phone network as fax tones.
I suppose it's possible that the cellular networks simply don't support this anymore, but my first guess would be that it's handsets that lack the software to do it, now.