Recently I discovered that the website used to make dining reservations at Walt Disney World opens at approximately 5:45 AM, not 6:00 AM as it had before.
I first discovered this while trying to make a dining reservation for Be Our Guest Restaurant 180 days in advance. I was online, logged in, and ready to go at 6:00 AM Eastern Time. But in less than a minute, Be Our Guest was already booked to capacity. While Be Our Guest is popular, it didn't seem right to me that it could have sold out so quickly. I later called Disney to ask if there was an error on the website. The booking agent explained to me that the site had opened to guests at 5:45 AM. So, basically I was 15 minutes late to the party.
I confirmed this on the Walt Disney World website as well. In the FAQ section, Disney explains that the website could open prior to 6 AM:
When making hard-to-get dining reservations, log into your My Disney Experience account a few minutes prior to 5:45 AM. I recommend testing your username and password the night before just to make sure you are able to log in. You don't want to discover that you forgot your password when it's game time. If you have trouble with your account, you can call 407-WDW-HELP and select option 3 for Disney Internet Help.
I also recommend using the website over the mobile app, but have both ready to go just in case. The website tends to be more stable and will operate a little faster than the app.