The following are possible points of failure:
- Your machine
- Your router
- Your ISP
- The Server's ISP
- The Server's router
- The Server machine
Connect to other servers. Do you experience the same problem on every server? If not, then the server has issues and there isn't much you can do about it. If so, continue to step 2.
Congratulations! You have already completed this step. You mentioned in the question that you have the same problem across multiple servers.
Host a server on another machine on your network and connect to it via LAN. Does it work? If so, your router, your machine, and you are not the problem, and you should blame your ISP. If not, continue to step 3.
Monitor the memory usage just before, during, and after you lose connection. Is the amount of memory used close to the amount of memory you have available? If so, then the game is stuttering because of constant memory swapping, and you need more RAM. I'm not sure if this can actually cause disconnections, but it can cause extreme lag. If not, continue to step 4.
Are you on a wireless connection? If so, find an Ethernet cable and plug directly into your router. Did that fix your problem? If so, then the problem is your wireless card. If not, continue to Step 5.
Buy a new router from a store that has a lenient return policy. Install it. Did that fix your problem? If so, then the problem is your router. If not, get your money back and continue to Step 6.
There is no step 6. I am out of ideas, but if you comment on the post with your extremely detailed results from following the steps above, I might suddenly have more ideas.