Are you trying to download new releases or really popular songs? There are lots of fake files around which will sit connecting forever.
Search again, look for artist/album rather than song title. Try the All Types search. Stay away from files that have huge numbers of sources (up to 100), go for files with 10 sources or less.

If you're finding that you still have this problem even when trying to download older files & regardless of source numbers... delete LW's preferences folder when LW's closed. See points 2 & 3 here
Fixes for LimeWire
Doing this will kill your settings within Options, you'll have to go back through & reset things afterwards