We've requested and gotten connecting rooms on 3 trips:
Trip 1: Me, DH, DS18, DS16, DS5, DD4
Trip 2: Me, DH, DS 23, DS21, DS10, DD9, DD4
Trip 3: Me, DH, DS12, DD11, DD6
Trips #1 & 3 were during the 3rd week in June (busy time) and #2 was the week leading up to Memorial Day weekend. We stayed at Pop Century on each of those trips. We chose Pop because it has the highest number of connecting rooms of all the resorts and we wanted to increase our odds of getting connecting rooms. Not having connecting rooms wouldn't have been a big deal on trips 1 & 2 (though we *really* wanted connecting rooms). Trip #3 would have been challenging if we didn't have connecting rooms since we only had the "littles" with us on that trip. Fortunately we always got connecting rooms and they were always ready when we arrived at the resort around noon.
When we went to Disney in 2013, we had 4 of the "kids" with us (ages 24, 14, 13, & 8) and we decided to stay in a suite at AoA and that worked out really well. We liked the extra space and having the larger fridge and a microwave. We're heading back in a few weeks and even though we only have the 3 "littles" traveling with us this time we're still booked in a suite so we don't have to worry if we get connecting rooms or not (plus, we liked the suite better than the 2 connecting rooms).
I will say that a friend of mine had requested connecting rooms at AKL several years ago (their trip was in late September). They had 4 adults and 2 young children in their party. They did not get the connecting rooms they requested. At check-in they asked if they would be able to move to connecting rooms the next day. They were told there wouldn't be connecting rooms available until day 4 of their 7-day stay. The CM was able to get them into rooms that were directly across the hall from each other, though, so that worked out ok for them.