I agree that if you're staying on Delta you are probably fine.
We had a problem last weekend when we had to switch from Delta to Comair which is a Delta connection airline.
We were delayed in Orlando because of lightning and then once we backed away from the gate we were delayed again because of bad weather in Cincinnati. When we arrived at CVG we had about 15 minutes to make our connecting flight on Comair.
The Delta rep upset us because we got off our plane and immediately asked the rep at the exit if we would be able to make our flight on Comair. He told us yes we would make it but we would have to hurry. Let me tell you, there is no way you can get from a Delta flight to board a Comair flight in 15 minutes. My little 5 year old was running as fast as she could but we just couldn't make it. The problem is that to get to Concourse C you have to go downstairs and get on a shuttle. This shuttle then brings you to the other concourse via a bus that goes among the planes between the gates.
We ended up missing our flight because of this shuttle and then ended up staying in Cincinnati that night because there were no other flights to MSP on a Saturday night. Delta treated us well with a shuttle to the hotel, one night stay, a travel kit, and meal vouchers but we still would have liked to have made it home that night. I realize that it wasn't Delta's fault that the weather was bad, but it made us mad that the rep at the gate saw we had 2 kids and told us to start running because we could get to gate C in time for our flight when because of the shuttle there is no way to make it.