It sounds like you are using iTunes Radio, and the songs only play for 30 seconds or so. I recommend starting by signing out of the iTunes Store:
Choose Store > Sign In (or Sign Out).
From: iTunes 11 for Mac: Manage your iTunes Store account
Then restart your computer:
- Choose Apple menu > Restart.
From: OS X Mountain Lion: Restart your computer
Then sign back in and test the iTunes Radio again.
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