I was delayed in LAX on the way to LAS due to secondary screening by border security (or whatever the tw@ts are called). LAX really is one of the nastiest airports for secondary screening if you are unlucky enough to be caught out by it.

Anyway, we just missed our connection as a result of the secondary screening. The AA agent (we were connecting with American Eagle) booked us on the next available flight out of LA which happened to be America West (just merging into US Airways at the time). The aircraft condition and service was repulsive, but the fact is that we arrived in LAX about an hour and a half later than planned and it's not much more than an hours' flight.

It really depends on the nature of the delay and the agent that you get. If it's border security, then I think the airlines are more flexible in terms of rebooking (so you could just lie!).