If you have done these then ignore. If I have a problem like this I ask the wife as being completely technology illiterate she will suggest some obvious things that we forget about. Doesn't always work but has helped.
May be worth looking at the Windows event log to see if anything happens at the same time you get disconnected. If you have reinstalled windows and the game there is a hint it may be hardware related.
Might be worth running a 3D benchmark whilst running the Windows resource manager and a temp monitor whilst playing to see if there are any spikes or other symptoms. Also, try running windows in safe mode. You can get it running with a network connection but it limits some of the other stuff it starts up with, may help rule out either windows or may point to a hardware failure.
Whilst it may be a hardware issue on the motherboard it could be fixable without going the whole way of replacing everything. A while back I was getting connection issues (generally, not just with BF4) and it was the motherboard ethernet port that was borked. Disabled the port and used a wifi dongle and all was OK, installed an Ethernet card and all was fine.
I would before splashing any money take the PC apart, add new thermal paste and gradually add components back. Add the minimum you need to play BF and see what happens. If you get the same problem, swap the drives to another port, use a different lead, memory to another socket and so on. That would hopefully indicate if a port or some other component was was failing.
It could be that a soldered joint is failing when it gets warm and they are really hard to isolate.
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