I ordered two items (a DVD player for my parents and a wallet for my Dad) at 2:00pm on Saturday and it showed the usual order in "X hours to get it by Sun 29 May". Chose One Day delivery. Received the email stating expected delivery 29 May and that's when the wallet arrived. The DVD, however, is due to be delivered today (30 May). I complained to Amazon via online chat and got a response that stated "One day delivery is fro the point at which we receive the item and then dispatch it. The One Day delivery is from the dispatch date". Given that the DVD was an Amazon warehouse deal I told them that didn't cut the mustard and got a month's free prime. I wasn't impressed at all though and it really felt like a corporate brush off.
At the end of the day though, Prime is still a bargain in my eyes. The TV is excellent (Bosch, Lucifer, loads of films), delivery is usually excellent, I haven't paid for an ebook in 2 years due to the amount of Super Saver credits I get, etc.