What kind of playlist are you trying to add songs to? Are you trying to add to a playlist in your library or one that's on an iDevice? In the former case you should be able to add songs to a regular playlist with this icon:

using either drag-and-drop or right-click > Add to Playlist. You can't use this method to add songs to a smart playlist, with this icon:

- in this case, you must edit the playlist's membership criteria to modify its contents. Nor can you add songs directly to a playlist folder with this icon:

- you must add songs to the playlists contained within the playlist.
if you are trying to add songs to a playlist on your iDevice, this should work as long as you've configured the device for manual music management. However, I'd always recommend not doing this - iTunes offers far more flexibility if you use playlists in your library to manage the content of your iDevices and then sync these.