When somebody has "pulled an Edward Cullen," they have acted creepy, stalkerish, or tries to break up with another person on the account of said dumpee being "too good for them."
Jenny: What? You're breaking up with me???
Pablo-the-Edward-Cullen-wannabee: Yes, Jenny... You're too good for me... You're like a saint compared to me.
Jenny: Really? I'm glad you're breaking up with me... Pansy. You just pulled an Edward
Monica: *is sleeping*
Edward-wannabee: * is staring at her*
Monica: *wakes up and sees him* WTF, you freak?!
Edward: *licks lips and jumps out window*
Monica: That freak just pulled an Edward.