On one of my very first OP orders, Fedex did a Saturday delivery when I expected it on Monday. I had not requested an early delivery. The package had a big sticker on it that said "URGENT: MEDICATION" and the driver said, by way of explanation, that that's why they bumped it forward a day.

That's the only time that's ever happened to me, but yes it can happen.

That's also the only time I've ever seen a package from an OP with an URGENT sticker like that. (The med was Viagra, if memory serves, so probably it was actually URGENT that I receive it early!)