If you're sure that you (or someone else) haven't already downloaded the files...

Which version of LW are you using? Which OS? Has this just started since upd@ting or upgrading LW?

Try going to tools > options > saving > advanced. Choose the 'always append (#) to the filename' option > apply > ok. Now, when you download the file, a number in brackets will show next to the file name (eg blahblah(1).mp3)...see if that at least lets you download files.

If that doesn't work, go back to tools > options > saving > advanced & try the 'always show the save as dialog' option.

If that's no help - are you using the default location for saving downloads? If you're using one of the latest LW versions, the default location is C:\Documents and Settings\your username\My Documents\LimeWire\Saved. Check your saving location by going to tools > options > saving...you'll see.
If you're using that default saving folder, try changing the location for saving downloads. You can change the location from that options page, just remember to click Apply to save any changes
See whether saving your downloads to a different folder makes things any better.

If that makes no difference, try deleting LW's preferences folder when LW's properly closed. See point 3 here & also point 2 if you've used older versions of LW in the past (there could be a folder in either location).
Fixes for LimeWire
This will change all of your settings within options, you'll have to go back through afterwards & reset things. If you're not sure of your settings, go through LW's tools > options & write them all down, so that you can reset things later.
You'll know if you've deleted the preferences folder properly...when you start LW, you'll go through the setup wizard again.

Vista users: LimeWire Pref Folder in Vista