I was listening to the radio this afternoon and Steve Wright (on Radio 2) and he played the song 'Sunzanne' by Leonard Cohen. I thought the song sounded very familiar but it took a bit of googling with the lyrics to find out why.
The lines (from Suzanne):
And you want to travel with her
And you want to travel blind
And you know that she will trust you
have the same tune and are very similar to:
You want to go out Friday
And you want to go forever.
You know that it sounds childish
from 'Hope' by REM. According to Wikipedia (so it must be true), the band didn't realise how similar the two songs were until after they had recorded theirs. As a result (and presumably to avoid and ugly legal action) they gave Cohen a co-writing credit.