I posted a similar question a few days ago hoping to get responses from people who maintain wiring regularly but no luck. I know that people do this and I've even seen it done but didn't think to pay attention back then.
I've done similar things myself replacing phone and thermostat wires for short runs. Sometimes it's easy sometimes it's difficult depending on what type of fasteners were used to hold the wiring down.
There are 4 different types of fasteners I've come across. This weird loopy thing that has a nail into the framing ring at the head, almost like a key ring but not as tight that allows the wire to slide back and forth easily. Then there's the insulated staple, wire tack, nail clip, whatever they're called that looks like a staple with a little strip of something on the crown. Those can sometimes be easy to pass wire through. Next is the plastic wire tacks and depending on the size/design can also allow cables to slip through. Finally there's the staple/double ended tack which if it's tight may not allow wire to go through.
Most of them don't dig into the wood deeply and on short runs you can loosen them enough to get cable to slip through them. The old 4 conductor wiring is pretty strong. Not sure about the 90s but the really old stuff was usually a thick gauge.
If you get someone at one end another at the other you can see if the wire is mobile. If so you can usually use that to help guide the new cable. If not, you can try to loosen the staples by pulling on the wire alternating directions on each end. Yank it a little from one end, then have the other person do it from the other end. Little by little you can sometimes manage to loosen or completely remove teh staples if it's not a very long run. It's time consuming but faster than cutting, patching and painting drywall.
The next tricky part is feeding the new cable. I've tried different ways of doing it including attaching the new wire to the old or attaching a fish tape to the end. You have to make sure you get a strong attachment that's not too much thicker than the original wire or it can get hung up and the splice will break leaving you with the end of the new wire in the wall. My fish tape is wider than the wire so I've had trouble using it through staples.
I've had better success splicing the new wire to the old. For Cat5e to 4 conductor telephone wire I would peel back the jacket exposing the individual wires so you have room to make a good splice. Be careful not to nick the wires while removing the outer jacket. Use the pull rip cord in the cable if equipped instead. Cut away 2 of the telephone wires and 2 of the pairs in the cat5e wire to reduce some of the bulk then splice those together and tape them up. My splices haven't always been great and I've been thinking of stripping the insulation on the wires to expose the copper, doing a lineman's splice then soldering the wires (how NASA splices wire) before taping it up for a better connection. I haven't actually done it this way yet but it should do a lot better than my previous methods of splicing.
Alternately you can use the poly line pull string. I know Klein and Greenlee both make them. They have them available in big buckets or small quart containers. Those can work but to get a good connection you have to tie the string over the jacket and tape it too. This adds some girth to the wire which may make it difficult to pull.
Hopefully everything works out fine but before you start anything come up with a backup plan in case something goes wrong. It's likely going to involve cutting into the drywall.