Been a member of Amazon Prime for over 3 years and I have never had any problems. In fact everything shipped directly from Amazon has arrived when it should or sooner. Have even dealt with returns and such and was very impressed with the customer service at Amazon, too. My guess is it was a fluke or you ordered from a seller within Amazon (but not directly from) who couldn't fulfill the request.
And lulz at you for blaming Amazon. The "one-day" option bumps your shipping priority to first, but it doesn't guarantee one day shipping. . . especially ON CHRISTMAS HOLIDAY.