I've never worked at a Disney Restaurant, but I did used to work at The Pirates League... if you showed up late then that became your new ADR time. For example:
You're ADR was at 7:30, but you showed up at 7:45. You're new ADR time is 7:45.
Let's say we're on about a 30 minute wait, you would probably get in around 8:15 (rather then 8:00). However, people can check in up to 20 minutes early, so imagine how many people with real 7:45 ADRs showed before you did, you would be sat after all of them (which will increase your wait again).
In most cases we almost always say the wait is longer than it really is. That's because there really is no way to estimate wait since it just matters how fast the person in front of you gets done and the chair/table gets cleaned. Also, Guests get angry if they wait longer then a "quoted" time and so we avoid that.
At our location we will always take someone after their ADR, but again it's not dining (we just use the same reservation system).