If it's a fault on the line such as damage, oxidation, being split between you and the house next door or the ringer wire being connected, then I dont think you'll see any improvement by switching to a new ISP.. It's like living in a farm down an old dirt track full of potholes, and ringing up Mitsubishi to say that your Shogun 4x4 has a very bumpy ride, and if they dont do something about it, youre going to buy a Range Rover instead.. Would you have them come and lay a new road to your farm?
Have you spoken to BT? Run line tests? Had an engineer visit?
Are you also aware that just because your neighbour gets a good speed, it isnt guaranteed that you will? You might not even be connected to the same exchange, or have the same distance to your exchange (wires dont run as the crow flies, remember)
Good luck with your new ISP though!