Poor BB Speed - possible faulty cabling?

Question

Hi Everyone,
I moved house 2 months ago and have had BB speed issues since moving - best speed between 200 and 500 kbps.
I live about 150 to 200 metres from the exchange, on the same road and can see it from by bedroom window - BT's website states an approximation on the speed of 21Mbps, some difference!!
My ISP won't pass the issue to Openreach as they state my BB connection is stable - they don't seem concerned about the speed, and I know if I report a line fault myself I am going to be charged £££ as the phone line is working ok.
I have already replaced the bottom half of my (very old and very yellow) NTE with an adsl filter one - BT fitted one of these in our previous house, that has made no difference, nor did removing the internal extensions and connecting the modem/router via a dangly filter to the test socket - I left it like this for a week.
I have been doing some investigations and the cable to our house from the pole (there is no green box on the road between my house and the exchange) seems to be shared with our neighbour - it's hooked up to the wall of the house near the gutter where there is a small black box underneath - 2 cables come out of this, one of them ours. Our cable is visually much older than the other neighbours - the neighbours cable is black and seems to be the same type of cable coming across from the pole, our cable is brittle and light grey and has 2 halfs, like very cheap bell or speaker wire (certinally not twisted pair by any imagination!) and runs right alongside the mains power coming into the house - both cables are running alongside each other for about 5 metres. This cable then comes down to the front door, and in behind the door fram - behind the door inside there is a very old looking bt connection box - brown in colour and oval in shape - in here is a rats nest of cables and what look like screw connectors and fuses!?! - the cable then leaves this (on much newer looking cable) along a short run of the skirting before going into the NTE in the kitchen, the modem/router is connected direct into the NTE (on the adsl port obviously!) 
I am convinced that it's this poor cable and the old joint box that is causing the issues? - How would I go about getting BT/Openreach to replace this if my ISP will not report it to them, and my phone line (for voice) is working?

Answer 1

'the cable to our house from the pole (there is no green box on the road between my house and the exchange) seems to be shared with our neighbour'
The cable from the pole to the property will either by a drop wire 10b, which has two pairs in it, i.e. enough wires for two line or a CAD55 cable, which is a four pair (4 lines).
'our cable is brittle and light grey and has 2 halfs, like very cheap bell or speaker wire'
That sounds like a drop wire 6, 6 as in its from the 1960's. Actually its not called drop wire 6 because of that but it would have been there since the late 60's or early 70's.
Anyhow, the only people who can replace that are Openreach and they have a policy of; 'if it ain't broke don't fix it'.