Okay guys, just use Rich's suggestion. But you don't have to go into a trash can.
Three of the songs on the album Heartthrob by Tegan and Sara only played the first few seconds. So, here are the steps I used to correct this problem.
1. Delete songs with this problem (Right-click song and click 'delete')
Unlike what Rich said, you don't have to find and remove the songs from your desktop entirely
2. Next, go to the iTunes store.
3. On the right-hand side of the iTunes store you will find a section called "Quick Links"
4. Choose the link "Purchased" (It should be the second to the top in the first column of the Quick Links section)
5. You will be sent to a page with a database on all the artists that you have purchased music from.
6. Find the artist that you are looking for, (in my case, Tegan and Sara) and click thier name. The artist names should be on a scroll-down list.
7. Click the artist or artists name.
Don't worry! Even after you delete a song from your iTunes library, it is always be in your Purchased database. (This only works with purchased songs)
8. All the songs you have ever bought from this artist will be listed here. All the songs currently in your iTunes library will have the word "Downloaded" next to their names. If they have been deleted, or have yet to be downloaded, they will have an icon with a cloud and an arrow. Click this icon on all the songs you have deleted previously and they will automatically start to download back into your library.
9. If the problem continues, repeat steps 1-8 or contact customer support
These steps will help you for items purchased directly off the iTunes store. If you have the same problem happen to songs that you downloaded off the internet for free or downloaded off CDs or other storage device, my guess would be to just try deleting and re-downloading them.
I hope I could be of some help!