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Can I convert this funny cable with 4 wires into an ethernet cable for connecting my laptop to the internet?

The cable currently has a white/yellow banded wire, a white/blue banded wire, an orange wire and a blue wire.

One of the ends is a BT connector and the other end is an RJ11 head with only 2 pins.

First of all: Yes you can, but it won't make you happy.

10Mbit/s Ethernet uses 2 pairs and works quite well over short distances with the kind of cable shown, you just need to crimp an RJ45 plug on it (using the pairs 1/2 and 3/6).

Now 10Mbit/s is not very fast, so you might want to go alternative routes. Assuming The idea behind using this cable is avoiding a recabling, you might want to consider Ethernet over powerlines, this works amazingly well in many home settings, the adapters (Devolo and TP-Link have worked for me) are cheap as chips.

EDIT

As requested in the comments, a few more details about the needed plugs:

Ethernet typically uses RJ45 plugs with 8 pins (1..8) carrying 4 pairs of wires. For historical reasons, these pairs are assigned to the pins this way: 1&2, 3&6, 4&5, 7&8. Older Ethernet connections use only the pairs 1&2 and 3&6.

So for a straight-through cable you need to connect pin 1 of one end with pin 1 on the other side, same with pins 2,3 and 6 and for a cross-over cable you need to connect 1-3, 2-6, 3-1, 6-2.

If the wires in the cable are paired (i.e. always two are twisted together inside the cable's protective mantle) honor this pairing when connecting the cables.

The mechanical process of putting the plug on the cable is called "crimping" and it is easily done with cheap "crimp pliers". If you don't have one, I recommend you just cut a cheap Ethernet patch cable in the middle and solder the two resulting "pigtail" half-cables to the ends of the old cable, using the orange/orange-white and the green/green-white pairs.

You should consult your Internet provider as to the proper patch cords to use when connecting to their internet ports.

Where I live, a phone line with an RJ 11 connector goes to a DSL modem and the DSL modem provides the Ethernet socket for an Ethernet cable to go to my laptop. RJ11 type phone lines do not directly connect to Ethernet ports, but of course BT may have different rules.

You can fabricate your own adaptor, but less headaches to find and buy the proper one.

I understand that im a little late to this thread, but there is a lot of misinformation on this thread. 2 twisted pairs does not get you 10 mb/s, it gets you 100 mb/s. Distance matters as much as a cat 5 cable would, there both twisted with rubber sheilding. I personally use cat 3 cable (phone cable) with my gaming pc. Cat 5 cable can transfer at a rate of one gigabit while cat 3 can transfer at a rate of 100 megabits. One gigabit=1/3 of a gigabyte while 1 megabit=1/3 of a megabyte. You will get decent speeds through cat 3 cable, which has 2 twisted pairs.