Connection new 4-line phones to an existing multiline network system
It would help to know what your old phone make and models were and what the models of the new phones you got are.

Based on what you provided so far I have a feeling your old system was a keyed system with a KSU or possibly a pbx and your new phones work without a KSU. The new phones are not compatible with your current phone system.

In a keyed system with KSU (this is also similar if you have a PBX as far as wiring goes) you're going to have some sort of box (KSU or PBX) mounted in a rack or more commonly on the wall for small systems. All the incoming lines will connect to this box.

All the phone lines that go to your phones will also be connected to your KSU. The phones usually only need one pair of wires going to them from the KSU to be able to access all incoming lines. The connection between a phone and a line is done through the KSU. The phones are digital phones that need to be compatible with the KSU you are using. Think of it sort of like a network switch or router for your phone system.

In a keyed system that doesn't use a KSU (like a lot of the panasonic phones that have individual line ports) all 4 incoming lines go to each phone. The signals going to your phones are just the regular analog phone lines. So each phone is basically directly connected to each incoming line. These are not digital phones and won't work with a KSU.

One thing to note, I simplified the connections but most likely you'll have a number of 66 blocks where the wires are actually terminated and not going directly into the KSU.

Update after phone models added:

Unfortunately I guessed correctly. The new phones won't work with your existing setup. You can either:


Get phones intended to work with you system (check model number of the KSU first). Even though Nortel went out of business you can still find used/refurbished phones for those systems. This is the easiest and cheapest option but you're stuck with an old system that is no longer made. All you have to do though is plug in the replacement phones.
Get a whole new system with new KSU or PBX and compatible phones. Get a similar system that can support your incoming lines and send one pair of wires to each phone. This should be pretty straight forward but you might need to hire someone to set it up. If there's any rewiring needed it should only be some minimal changes at the telecom closet. I haven't used it but the XBlue X16 system seems very popular and should work with your setup with maybe some minor rewiring of cross connects at the 66 block. 
Get rid of your old system and use the new phones on their own. This is going to require that you have 4 pair wire going from the telecom closet to each phone jack. If you already do have 4 pair (8 wires in the cable) going to each jack then with some minor changes you can use the new phones. If you don't have 4 pair cable going to each jack you're going to have to run new wires which can be expensive.