In 10BASE-T and 100BASE-TX, one pair of wires is used for transmitting, and one for receiving. That is, one pair is the pair the Ethernet host transmits on, and the hub or switch receives on, and the other pair is the pair that the the hub/switch transmits on, and the Ethernet host receives on.
If you split the cable with a simple passive splitter, you're hooking up those two Ethernet hosts transmitter-to-transmitter and receiver-to-receiver. That's like holding the phone handset upside down and trying to speak into the speaker and listen to the microphone—it just doesn't work. So even if both were in half-duplex mode (like they were hooked to a hub, not a switch), neither of the Ethernet hosts would be able to sense when the other was transmitting, because neither one's receiver was hooked up to the other one's transmitter. So they would have undetectable collisions. Not to mention that they'd both be connected to the same port of the hub, probably confusing the hub's autonegotiation ability, because hubs don't expect to autonegotiate with two separate hosts on the same port.
In many ways, things are even worse in your case of hooking them both up to a switch, because they could both end up thinking they can do full-duplex, which means even more undetectable collisions, on what's supposed to be a collision-free link (properly-wired full duplex links can't possibly have collisions).
With 1000BASE-T (Gigabit Ethernet over Cat5 or better UTP copper cabling), the situation is even worse, because all 4 pairs of wires are used for both transmit and receive (simultaneous, full-duplex), and the transceivers are sophisticated enough to enable that. But if you suddenly have a third party on the line transmitting and receiving all at the same time, it completely blows away the way the simultaneous bidirectional signaling scheme works. With three devices all transmitting at the same time, even when you subtract out your own transmission, you can't differentiate the other two devices' transmissions in the signal you're receiving.
Some early flavors of Ethernet, such as 10BASE-2 a.k.a. "thinnet" a.k.a "cheapernet", featured a bus topology where all the hosts on the LAN literally shared the same wire (the same coaxial cable). Because the same wire was used for both Tx and Rx and there could be any number of hosts on the bus, it had to be half-duplex. But a 10BASE-2 transceiver was expecting it to be that way. And since all the transmitters and receivers were hooked up to the same wire, everyone could hear each other (unlike your split 10/100/1000BASE-T example).
First of all, you can only have one modem/router on a single phone line. So if your jack in the bedroom is hooked up to the same phone line as the one in the living room then you can't get a new modem/router to have in your room.
You say there are two jacks in the living room. If one of them is used for hooking up the modem/router then what is the other one used for? Could that perhaps be connected to the jack in your bedroom? You can find that out by getting a phone line tracer (device that sends out sine waves over copper wires and its corresponding receiver) hook that up to your bedroom jack and take the receiver to the living room and see if it sees the signal. If you don't hear the signal there, then that means that the copper line connected into your bedroom has no connection with the copper lines connected in your living room and there is no way to get any signal between the two places using the preinstalled copper lines.
If you do hear the signal in the living room that means the jack in your bedroom is hooked up to the same line as the one in the living room.
Then there are two options.
This is all the same line, the line that goes into your current modem/router, the second plug in the living room and the plug in your bedroom. Then you can't use that line to transmit any extra data.
The second plug in the living room is connected with your bedroom plug, but the plug which the modem/router hookes up with is not connected to that same line. Then if there are 4 wires between the living room and your bedroom you are in luck. An internet cable really only needs 4 wires, which is what the RJ11 plugs (I am assuming here that the plugs in the living room and your bedroom are RJ11 plugs, look them up if your not sure) are. So you can create or have someone create two cables with RJ11 on one end and RJ45 on the second end. Hook one of them up to the modem/router and to the living room plug the other one would then go to your bedroom plug and to your computer. Voila, you got fast internet. On the other hand, if there are only two wires connected between your bedroom and living room, your out of luck.
So basically to use the plug in your bedroom you need four wires in that plug that connects to the empty plug in the living room and those wires can not be connected to the plug that your router connects to.
A second solution to get net to your bedroom is using the electricity wiring. See for example this cnet article for options of makes and modules to do this.
Best Answer
You could look into a wireless bridge (or WiFi bridge). This may be the component/term you're looking for in your situation. It's basically a reversed access point and it's a feature some WiFi routers have built-in. You connect it to an existing wireless network and it then "outputs" the network on its ethernet port or ports, usually acting as a switch depending on how you configure it.
A second option might be powerline networking. It's a technology that uses existing powerlines in your home (yes, using the sockets you plug your toaster or microwave in) to piggy back network signals. However, this is very dependent on how your home is wired and your mileage may vary in terms of speed, latency and stability.