Actually, it doesn't matter if the flights are on the same or different carriers: as long as both flights are booked on the same ticket, and the connection time is legal per the carrier rules at that airport, you're protected from a mis-connect. If you miss the connection, then the carriers involved will collectively work to get you to your ticketed destination as a standby passenger on the next available flight with available space, without extra fare or service charge to you.

15 minutes really isn't enough in every case. Because of the "15-minute rule" you need to actually be at the gate of the connecting flight 15 minutes before published departure time or you risk losing your seat (this doesn't always happen, but it can). As long as you don't do something stupid like sit down for a meal in an airport restaurant, if you're unable to make the connecting flight using a reasonable and normal pace going between gates, they'll take care of you at no cost if you miss the connection.

By "take care of you" I mean that they'll a) confirm you on the next flight with an open seat in your same cabin regardless of fare class, and b) if there are flights before that flight, allow you to stand by for those flights to see if a seat opens up at the last minute. Note that the "next confirmable flight" might not be for awhile, and in some cases not even that day or in the worst case, could be several days later. If that happens, you can stand by for each and every flight in the meantime, and usually you'll be able to get on something. Other alternatives which you may have to suggest is to use alternate routings (that is, take a connecting flight instead of waiting for a non-stop or direct), or in the case of multiple carriers, see if the carrier operating the first leg can help you.

In my experience, when you miss a connection on a single-ticket itinerary, the carriers will go out of their way to accommodate you on their own flights without trying to upsell you or otherwise make life difficult. But they have only a limited number of seats on each flight, and if flights are full, there's nothing you can do but wait.

Also, if you have checked baggage, do not expect to see it until you reach your final destination. There's a chance that it will make your originally-scheduled connection even though you don't, and if not, most likely it will be put on the next flight to your destination, even if you're not on it. It's very unlikely for a domestic flight that it will stay at the connecting airport with you, and it's very unlikely that you'll be able to retrieve it at the connecting airport.