Almost on our way, again

Our plane was 15 minutes delayed taking off because Delta booked multiple people to a seat and they had everyone running around trying to get a seat. The we were another 5 minutes delayed pulling in on landing because another Delta plane was in our gate. Then the our connecting flight closed their gate at least 15 minutes before take off. So, our 45 minutes to get from one flight to the other quickly became 10. We didn't make it. Delta says it was our fault. So we had to pay for an uncomfortable hotel and have no luggage and no clean clothes. I called Delta about what will be happening to our luggage. It would seem that even though they didn't have time to let us on the plane, even though we made it before the plane was ready to leave the gate and before our departure time, they did manage to have time to remove our luggage so it will be on the same flight as us this morning.