I’m trying to pull a new thermostat wire down from a first floor room down into the basement. The exit point in the basement is in the ceiling, between two ceiling boards pretty inaccessible. Would it be stupid to try to hook some fishing wire to the old wire, pull that through, then tie the new wire to the fishing wire and pull that through? Is that unlikely to work? I don’t know about the opening diameter, and I’d be going from a 2-wire setup to a 4 or 5, so I guess it’d be thicker.
Your odds of success are about 1% unless the wire in in conduit, and it almost certainly isn't.
Generally, wires in walls are stapled to the framing. Perhaps your thermostat wire isn't, and then you might have a chance, if it's a straight shot. If there are staples or the wire goes around corners, you probably won't be able to pull the old wire out, and if you do manage that, the point where the new wire is connected to the pulling wire will not be happy about going through staples or around corners.
Posting by request: I tried this, and it worked with ease. I severed the old cable at the furnace, and joined it firmly to the new cable by twisting the wires and then carefully wrapping the connection with electrical tape. I made sure the tape tapered off smoothly on either side of the join, so as to make the overall effect smooth. I then went to my thermostat, and pulled out the old wire until the join and new wire came through. There was hardly any resistance. I assume if your wire is stapled this wouldn’t work for you, but that wasn’t my case fortunately.