I had this on last Thursday. My flight on UA to SFO from LHR was late. I got a text from them just before I went to bed, so could set my alarm for later. But I knew that the plane was coming from SFO and it couldn't possibly arrive any earlier. But UA had options: they could have cancelled this, their second and last SFO flight of the day, delayed their first and put everyone on it, then kept the plane in London, cancelling an inbound for the following day. But I checked the loads and saw that this was not really feasible. So I was safe. The moral is that you should only turn up late if you really see that the airline has no real options but to fly the original flight late.