The same thing happens to me,the problem is with the program (a major bug).
I run a 10Mb Broadband connection and every minute I was getting a message about no network connection...I have a permanent net connection for goodness sake. I went and downloaded a 3.5GB program using flashget at 1200KB without once having connection problem.
I know Limewire is the problem because even if I am downloading a small file I cannot access the internet and I am running a dual core 2.8ghz processor with 1GB of DDR2 ram.
I even went and tested another P2P program and the result was no problem.
I finally fixed the problem with Limewire...I uninstalled it as it is a major pain!