Gillian--
We DID hear some noise through the door, but mostly at the expected times--when we were moving in, when we were coming back from the shows at night, that kind of thing. I only remember hearing their TV once.
Our experience is that the noise problem is the opposite wall--the wall that the beds run along. The beds in the adjacent cabin also run along that wall, and so from 10:30 pm to 9:00 am or so--overnight--you've got most of the activity in the two adjacent cabins along that wall. (Snoring, baby's crying, clonking against the wall in the middle of the night, etc....)
We were in 6538, with a connector to 6536. The family next to us in 6540 (part of our group--my business partner's family) have a small baby--Dave.
We heard way more from Dave than from the folks in 6536, and that was during the overnight period. (Of course, we love Dave, so we just ignored it...)
I guess the best course of action is to get a room with no connecting door, and hope for quiet neighbors.
We've noticed that if we are reasonably quiet, our neighbors seem to get quiet. We don't have the TV on much, and if it is, it's not on loud. We don't make a lot of noise in our stateroom, just because we know we have close neighbors. It seems like people figure it out, and respond in kind after a day or so. . .
On an earlier cruise we were neighbors to a couple who were dedicated to various hijinks on their verandah. We were also next to a retired couple once, and the gentleman (nice man, as I recall) used the verandah for smoking his cigar.
Both of those situations were a bigger annoyance than the noise through the door or from baby Dave.
RDT