It sounds like the ID3 tag (part of the MP3 that stores the song name, artist, etc.) is using an old version (v1.1), which only supports around 40 characters for each field. To check the tag version, right-click on the song and select "Get Info". On the Summary tab, there is a field "ID3 Tag:" with the version number.
In addition, it sounds like you have iTunes set to automatically organize your music. When this option is set, iTunes renames the songs according to the tag. So because the tag only has 40 characters, it only uses 40 characters for the filename.
You can use iTunes to convert tags from 1.1 to 2.3 or 2.4 (I suggest 2.3 because not all applications support 2.4), but that won't fix the short file names.
If you haven't added all of your files to iTunes yet, you can fix the tags using an external MP3 tag editor like this one:
http://mp3tag.de/
Unfortunately for the ones that are already in iTunes, you pretty much have to manually fix them. That is, unless you can find another music program that can download track info from the internet and automatically tag your files with the correct names. I don't know of one off the top of my head.
