We were able to, for a $10 change fee (don't ask me where that number came from). We were slated to start our 4/3 trip at AKL on Sunday, May 3 but moved it to start on Monday, May 4. We called up and made the reservation change 3 days before check-in. I was fully expecting to be told 'no' or to at least have to pay a $50 change fee, but was incredibly thankful that we were able to move our dates due to a sick pet.
I think it depends on where you are staying, what the availability is, and what CM you speak with on the phone. A friend of mine tried to move her 4/3 reservation as well. She was staying at the GF and wanted to move it by 2 days, but was told that there was no availability unless she wanted to pay full price.
Good luck.