Can you connect another computer to the modem? , if so does the internet work OK?.
It appears by your letter that BT is aware of the problem, but is slow in finding the problem- How long have you had the problem?.
If this is a wireless connection, could you fit a cable between the modem and the computer to try out a direct link.
I'm wonndering if the battery in your laptop has developed a fault and is no longer supplying enough power to keep your system setting OK-- Does it work alright when plugged into the mains?.
Just suggestions to try