I've done it on a Delta flight with 2 adults and 3 kids...but like PPs have said, it isn't fun. Bring as few carryons as possible, especially if you will have little ones in tow. We skipped gate checking the stroller specifically because our connection times were so short. We did arrive about 10 minutes late, but it was announced that for those of us with short connections, the outgoing flights would be held for an extra 10 minutes. We were among the last to board.
Are you stuck flying with Delta? Other airlines (American specifically, IME) are much better about accommodating you on alternate flights if you miss a connection for reasons out of your control.
If you do miss the connection, don't expect much out of any airline as far as lodgings, etc. Since late flights are often blamed on weather or "operational" factors (runway not open at the airport, etc etc), and those reasons are specifically excluded from the reasons they deem their responsibility, you will often get nothing more than a "Sorry for your inconvenience" and a suggestion of a nearby hotel.