It is still broken.
Yea, my money is on your router. But here's 100 questions. Is your system set to use a static IP or dynamic? Any other computers in the house? They have the same symptoms? If no other computers or they don't have the problems, is your connection wired or wireless? If it's wired does your system have an ethernet expansion card or is it built into the motherboard? (if the place the wire goes is up in the same area as most of the usb slots on the back, it likely is built into the motherboard) If it's an expansion card do you know if the shops tried another card? If it's built in any capacitors look odd (seem to be expanding upward or have a brown substance coming out of them) or any burn marks?
If it's wireless are there any thick concrete or metal based walls between your system and the router? Are there any other wireless networks nearby using the same channel? Wireless card or built in...
There are a ton of things that can happen to cause intermittent connections. Surprises me that they never managed to fix it though, after the 2nd visit I would think they would consiter possible router issues. Be an good excuse to charge for a housecall at least. But as I said, if you brought the computer in a number of times and it hasn't been solved the problem might not be the computer.
By the way I don't expect you to answer those questions. Those are just the questions I would be asking if I was a tech trying to work on the system. Though most of those I would answer myself after a look see.
Wow 8 ninjas while I was writing this.
nd it's not related to the router, because it happens in other places, and even when i'm not actually CONNECTED to the internet
Hmm. bad cabling maybe? If they replaced the system board it's probably not that, unless it's a bad part that fries other parts... Might still be Operating system related. System Restore usually works, but I've seen situations where system restore SHOULD have fixed the problem and hasn't. At that point I would be thinking complete OS reinstall. Maybe trying a repair first if possible, see if windows notices any bad files. But of course an OS reinstall is pretty drastic. And it might be better just to live with the problem than have to back up all the data and hassle with reinstalling software.