This is true. If it's all one airline, then they take responsibility for either delaying the plane, or getting you on the next flight plus will put you up for the night and pay for your meals if neccesary. In fact, they may do this even if it's not all with one airline - as long as it's one they've got a partnership, code share or special agreement with.
I was set to fly once from San Francisco to Honolulu with United Airlines, then on to Hilo on the Big Island. But when we boarded at SFO, the pilot noticed something wrong with the plane (the infamous "light" in the cockpit) and we all had to disembark again. The problem was though that as the crew could only fly on a 747, we had to wait for a new one to fly in from elsewhere in the States (they only had a DC-10 spare at the airport I believe) and this wouldn't arrive until 6pm - seven hours after we were meant to take off. It also meant that I would miss my connecting flight as Hilo airport closes at 8pm.
They gave me two meal vouchers (lunch and dinner) that I could use at SF airport, booked me on the first flight the next morning to Hilo (5:30am!!!) and arranged for me to stay the night at one of the Honolulu airport hotels, plus paid for the shuttle bus to and from the hotel (admittedly there were about fifty people in the same boat). They would also have paid for breakfast, but as no restaurants at Honolulu airport opened until 6am, that wouldn't have been much use!
And the connecting flight I'd missed wasn't even with United, but with Aloha Airlines.