[EDIT] 3.6.15 is live and will automatically take care of these situations for you. [END EDIT]

This will happen if you first installed 10.0, then upgraded to 10.2 without installing some specific packages. We are in the process of revising our installer to fix this (and some other) issues. In the meantime, you can continue running LimeWire one of two ways.

The best way, if you still have the OS X installation CDs is to pop the first CD in and navigate to:
Mac OS X Install Disc 1 -> System -> Installation -> Packages -> BSD.pkg
and double-click to install that package.

Alternatively, you can go to /Applications/LimeWire, ctrl-click on the LimeWire application, choose 'Show Package Contents', open on the 'Contents' folders, ctrl-click on the 'Info.plist' file, choose 'Open With', choose 'Other', scroll down on the right and choose 'TextEdit". Within TextEdit, find the line that reads "<string>prelude</string>" and change it to "<string>LimeWire</string>". Save the file, close TextEdit.

After you've done either procedure, LimeWire should open as normal. We'll be releasing a new installer soon so that all this isn't necessary if the user didn't have BSD installed, but for the existing installations these steps are required.

Thanks,
Sam