There are a couple of things to remember. The max distance for Cat5 is 100 meters, but that isn't the length of the overall cable, it's for the wires inside which are twisted. This reduces the overall length you can run the cable. You should still be fine with 240 feet though.
Also when connecting a switch to a hub you use a straight-through cable, but when connecting a switch to a switch you need to use a crossover cable. Some switches autosense crossover status and will work either way, but it is always better and will prevent random problems if it is cabled correctly.
The best option for troubleshooting is to use a cable tester on the 240' length, but if one is not available you can try to hook up a computer directly to the end of the 240' length of cable and see if you can get a connection.
If your cable is a straight-through you should be able to get a crossover converter from a local store if you do not feel like running another cable.