The roof on my convertible will only open to half-way stage. BMW claim a cable must be replaced at cost of £750. Is this a reasonable estimate?

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try a spray lubricant into the motor mechanism and grease any visable wire cables that run from/to motor ,not the electrical cables the bare wire winder cables,also grease/ lubricate any accessable pivot/hinge points , gently pull the cord and gently push the roof back and forth occasionally trying motor with a bit of patience this should get the roof moving , also look for anything that may be snagging the roof/mechanism as there is a pressure protection system to stop the roof being fully closed on say someones hand so preventing injury

I am not sure what you are trying to do. But if the problem is that the roof will only open to the first stage, sun roof, and no further, the problem is the package shelf in the trunk.

If the package lid gets pushed up and is not pushed back to it's normal position, the roof will fail to open completely. If you lift the bottom of the roof at the boot, there are levers on either side to release the bottom edge of the roof, you can make certain that the package shelf is seated correctly. Once the shelf is seated, the roof will open in all of it's glory.

check the parcel shelf has been refitted correctly as there is a micro switch on the left hand side that stops the second stage of the roof from opening

Check the cable to see if it's not broken

i can,t see why all these should need to be replaced,you need to get a more detailed report of what exactly they want to replace and for what reason in my experience dealerships dealing with roof faults will replace parts one by one untill the problem is fixed and surely the dealer should cover the cost of the repair if the roof faulted after only 3 weeks

anything stopping it from opening up?

Ha, i just replaced one of these today. Yes they are serviceable but a pita. There is also a fix for the rear glass not closing. The rear tilt mechanisms need to be replaced bmw part number 54 13 7 240 682.

The Roof Mechanism works with two motors similiar to the ones that wund the windows. Now the two mechanism are located in the left side of the trunk. 

The trunk wont open Because of the roof lid that is not closed properly. How you can test this is to press down on the right sid of the roof lid and then try to opne the trunk.

Now the reason that this is happening is that the roof lid is not closing properly nad pressing on a small orange buttin located just below the right hand side roof lid catch. How you are going to remedy this problem is to adjust the cable tension to close more securely.

Now the two motors in the trunk that works the convertable roof can be engaged and disengaged on motor works the roof lid and the other one the roof.

You need to get these motors in sync so that they open and close the roof in the correct order.

That is if the electric are still working.

If you open the roof lid and look at the lid catch on button side all you have to do is adjust the cable tighter so that when the catch goes down it engages the button. You can do this by turning the nut on the adjuster. The adjuster is located just under the catch on the cable leadeing to the motors