Id still recommend sourcing the exact issue as it may be your hardware (router/pc/laptop/wireless) thats causing the issue. What happens if you move to NTL and still have the same problem.
When BT are saying its the line, have they said for sure that they have encountered, and can see there was, an issue with your line? They will have monitoring software for this.
Are you experiencing the drop on more then one client (pc/laptop) is it wired or wireless inside the house? If wireless, can you use your intel config software on your client to see what the transfer rate to your router is at time of drop to rule that out?
If thats ok, can you go on to your routers config page to see if its dropped the IP?