Why don't you grab a second modem from eBay and unlock it if you don't want to unlock yours.
Im with Zen and there was a major problem with my line due to a faulty cable joint under the pavement. The graphs you get with an unlocked modem give lots of information, I could see noise increasing on my line at random points throughout the day.
When I reported the case to Zen they were more than helpful, they didn't give up with BT. In the end openreach dug up the road and repaired the cable.
Better still I even managed to wangle getting my line fed from a different set of D side cables, just by chance my house was at the end of two d side loops that run either way around the estate. In the end phone phone line was a couple of hundred metres shorter and my connection was even better than when the fault first occurred.
Freeserve Dial-Up --> BTopenworld --> <n>ildram -->Talk Talk LLU --> ZeN