This might (just might) be a solution to the connectivity issue (and ONLY the connectivity issue):

Get the following programs:
..1> The latest version of LimeWire. Get yourself the latest version today.
..2> The latest version of Java (that's Java SE v1.5)

Get the following updates:
..3> Update your Windows OS with the latest updates.
..4> Update your antivirus/firewalls with it's latest updates.
..5> Reboot your PC after the updates. Don't omit this step.

Uninstall the following:
..6> Completely uninstall LimeWire. Delete the installation folder too! (Just to clean up clutter..)
..7> Del. it's Settings folder manually. It's a folder called .limewire in C:\Documents & Settings\(user)
..8> Uninstall Java completely from your system.
..9> Reboot your PC after the uninstallation. Don't omit this step.

Install the following:
..10> Install Java first, before you install LimeWire.
..11> Turn off Windows firewall (or whichever firewall you're using) - or if you know what you're doing, confirgure it correctly.
..12> Install LimeWire.

This might solve your connectivity problem. Note this is only a possible solution - Not THE absolute solution for every computer in this universe. Worked for me, and I hope it works for you too.
And it's only expected to solve your connectivity issue - you may still have to deal with download issues.

Let us know if this helped. Post back. Please do! With whatever suggestions you may have.

Regards, AG