Ways it could work, in theory
Page data could be a vector, however any receiving fax is required to count lines of wrong lengths as "errors" and discard them. The number of lines per page is of course not limited, but faxes have been used for long pages in the past.
What's actually more likely is the negotiation phase, there Fax machines talk HDLC over v.21 with eachother. It could be that the software only allocates a 256 octet space, but the HDLC frame is much larger. Since HDLC frames have no "length" indication you don't know how long they are. I have actually seen one particular modem (ELSA Microlink 56k) crash when it receives bad negotiation in that phase.