My Sister had a similar 'problem' with her system ... I was convinced it was BTs std modem, got a replacement, wired it up using existing cables, there was no difference in the cables, only to find, oh well, we got a spare modem.
Tried replacing the cables, just in case there was a difference, but noticed there was no 'flickering' light on the the Ethernet port. This was easily remedied with a PCI Ethernet card. Cheap as chips, worth a shot.
It could be another 'problem' so other advice is more than welcome.