Well...anytime that you are connected to the Internet...you compound the possible issues.
Could be your connection, could be something encountered on the Internet...could be your network device, PSU, etc.
If the system functions well without connection to the Internet...it's not the system.
The best thing to do would be to look for clues
That means looking at Event Viewer (or Win 7 equivalent) and looking for .dmp files that might result from a BSOD.
How To Use Event Viewer - http://www.bleepingcomputer.com/forums/t/40108/how-to-use-event-viewer/
Help Diagnosing BSODs And Crashes (BC) - http://www.bleepingcomputer.com/forums/t/176011/how-to-receive-help-diagnosing-blue-screens-and-windows-crashes/
When dealing with Internet issues...system power is not the major consideration, it would be bandwidth (I think).
If System A is twice as powerful as System B...but has the same upload/download capabilities...System A will not function any better or worse than System B, since accessing files would be the only other variable (which is governed somewhat by the hard drives). Yes, a faster CPU theoretically can access files in a speedier manner...but there's a limit to how much of that is CPU versus RAM versus hard drive access times.
Be that as it may...the differences between an I5 and I7 system I would not expect to be dramatic.
I don't think there's an easy answer to what you are asking...but I've been wrong before (many times)