I always think of the cutoff times as a promise they will hold my seat until that time, but after that all I can do is hope they are able to hold it, but there isn't a guarantee.

Things that could affect whether or not they will hold:

1. number of standbys/oversolds
2. connecting vs. originating - if you are a connecting passenger, they know what time your connecting flight arrived and the gate you are coming from (and if they estimate from the knowledge that you won't make it, they aren't going to wait even then). If you are originating, they have no idea if you are stuck at the end of a long security line or just in line to use the bathroom two gates down.
3. weather - if the captain is telling them the flight has to go or the entire flight is cancelled, they aren't holding the plane. Conversely if there are favorable winds, they might be able to be more generous.
4. Whether the gate is needed for another flight

And the list goes on. Basically do everything you can to be there by the 15/30 minute deadline, or risk losing your flight.