There are a few things to check here.
- Check the integrity of the cable by connecting that same cable to another laptop (maybe a friends) or to a desktop PC (if you have one). I know you mentioned this in your post, but it's good to double and triple check.
- Another check would be to connect that same patch cable from your modem to your switch/router/wap. If you lose connectivity to the external network (WWW), then you can determine the cable is compromised. As you said,
Strangely enough, the other day, it suddenly did connect via cable, me not having done anything. Then the next day it was back to "normal" again.
- Another possibility is that the RJ45 port on your laptop is not functioning correctly. If you have any Windows partition on the laptop, you can try booting into Windows and reconnect the patch cable to the Rj45 port and see if it connects to your network immediately and gives you internet access.
When the patch cable is plugged in, use
ifconfig from the terminal to make sure the eth0 is actually being seen. Also try
ifup eth0 from a terminal to turn on the local ethernet interface.
I ran Linux Mint 17.1 Xfce x86 on my Lenovo Y410P for a number of months with no issue connecting to my network via the RJ45 port.
In your case, if you have previously had Mint connecting to the world wide web from your notebook via patch cable, the troubleshooting should be relatively easy from here.
I know my answer probably seems redundant, but layer 1 in the OSI model is the physical layer and the beginning place for all troubleshooting.