This actually happened to me last month. I was connecting in AMS from JFK to IST.

My original routing left 1:20 for the connection. I knew that the next flight was 11 hours later - 8pm.

My outbound from JFK was delayed more than an hour and I started to think it through and decided that I would be happy to have a full day to hang around Amsterdam.

As we were landing, it was actually close and I was imagining what would have happened if I had come in, say, 20 mins before my connection time and I simply decided to walk slowly.

Who's to say how quickly you can get off a plane, through a terminal and get to another flight?

I pictured ambling slowly up to the transfer desk and telling them I had been delayed and missed my flight and needed to be rebooked - as long as the original flight was really delayed and it was close, what can they say?

As it turns out, we were a good bit later than that so I had no need to fudge it, but why not.

The downside on this type of thing is that wandering around on arrival in Europe on very little sleep is never ideal, but what the heck. I had a nice day, did a little shopping and got some sleep on the flight that got into IST at midnight.