Just make sure your story is good. I've heard quite a few and they'll bring a manager out, but not every case, even if they are 100% true will get you an exception.
One case, there was a family with a small emergency that required them to go back to their hotel. As a result, they missed their Fastpass time. Through their story they mentioned vaguly that they got back to the park just a couple hours later than their Fastpass, but rather than go immediately to it, they waited until the very end of the day to use it like they usually do. Final result was no admittance.
Another was dinner reservations at some restaurant at 4pm. Turns out the restaurant doesn't have reservations during that time. No admittance.
Moral, if they catch you in a lie, find inconsistencies to your story, or you abuse the late rule to an extreme they will still deny you. If you are late, you should still show up as soon as you can. Other cases I have seen managers give in slightly by writing them out a Fastpass to return but based on the current wait time allowing them to have a new Fastpass and allowing them to go do something else, but they still have to wait some time. Oh, and if you knew you had a reservation during the current distribution time, then don't get a Fastpass at that time. If you knew you wouldn't be able to make it, then they don't have to honor it being late either.
The 5 minute rule is a the discretion of the cast member based on current operations.