We did this. My daughter got very very sick on a 7 day cruise, and the Ship dr. advised that she NOT fly home - so, we booked a second cruise and did a back to back - we booked the day before the 1st was over.
It was not an ideal situation, however, she did recover, and we were able to enjoy ourselves. IT worked for us, as we had any food we needed, the Dr was 5 mins away (she was seeing him 3-4 times a day) and there was entertainment if we wanted it. The kids club was awesome in that they created her a "play package" she got some playdough, coloring books and crayons, markers, stickers - all to do in the room while she was so sick. It was really nice of them.
I truly think a DCL cruise is the only place you will hear someone say "my pleasure" as they clean up your childs throw up.
So, yes, you CAN book the following cruise while you are on-board, if there is room. We did it.
Lizzy'sMom