cat5e is the same as cat5 but is rated for 1 gigabit. It is impossible for a rounter to tell wether a cable is cat5 or cat5e at 100 Mbps speeds. In all likelyhood you have a bum cable or bum RJ45 connectors on it.
#1 Test the router with a short cable that you know works. You may be blaming the cable when the router or the NIC/PC are at fault.
#2 Assuming #1 worked, you know you have bum cable. If you are able, redo the ends and try it again. If you are not able or don't have the rather expensive crimper and or know-how to do this, take the cable back to the store for a replacement. It is not uncommon to buy bum cabling. Also, you may have a piece of patch cabling designed to connect a router to a hub. This does not work as a standard ethernet cable (although the ends can be removed, and it can be wired as a standard cable.
That should get you started on the troubleshooting.
What follows is not directly related to your question, but you may find helpful:
I wired my whole house for gigabit ethernet this summer. I ran cat5e into a jack in the wall in every room. I made all my own ends and wall jacks. This is time consuming and a bit more expensive, but not all that hard if you have the time and the tools. Home depot sells cat5e by the foot. So does Menards. They both also sell the wall jack plates, etc. the computer geeks have a good price right now on some of this stuff if you are interested.
Here
Best of luck,
James