I'm going to assume you're using Windows & LW 5. If not - please tell us your OS & LW version

Is iTunes set as your default player?

LW 5 has the ability to automatically add files to iTunes (which LW 4 never did for Windows). If you've got LW set to automatically add files to iTunes, then once a download's finished iTunes opens & the song's added. But it doesn't automatically start playing in iTunes.
Have you sorted songs by date added, to double check that new downloads aren't there? In iTunes, right click on the column header & tick 'Date Added'. Double clicking on the Date Added header will let you show songs in whatever order you like - either most recently added at the top, or first added at the top

Go to My Library > right click on a song that won't play > locate on disk. Once your saved folder's open, right click on the file > open with > choose a different player. Any luck doing it that way?

Sometimes LW has a glitch with adding songs automatically to iTunes. In LW go tools > options > downloads. Untick the box to automatically add downloads to iTunes > ok. Now go back to tools > options > downloads & tick the box to automatically add to iTunes. Sometimes the feature to add files automatically can fail & you need to disable that option & then re-enable it.

And if songs won't play with any player, you might've downloaded bad files. Check out these links for more info.
Virus thread: Typical virus spam file sizes - Make a note of them (click on blue link to see)

* WARNING: Viruses on network you should be aware of ! *

Fake files showing up in search results

How to find music