On our last cruise, we were surprised at how much room there really is to store things - way more than we needed. There is plenty of closet, drawer and shelf space for clothing. If you have a sofa, there may even be drawers in them as well. As for bags, under the bed is a good option - we just stood ours up on the floor of the tall closet with no problems.
That being said, you don't want to overpack - if you are like me, you won't use half the stuff you bring anyway.
When I looked up the definition of "over packing" there was a picture of my wife. Still, we always find room for storage. We empty our FOUR suite cases, and 2 carry-ons, and put the smaller suit cases inside the larger. Every thing fits under the bed. On our coming up back to back cruise she's going to try packing only 3.
We always seemed to have four checked bags, and one is right up there as far as the airline limits for size. On our last Alaskan cruise I insisted on downsizing to three pieces. Given the weight of the bags, we still had enough stuff for four suitcases but we had 1st class air so the weight limit was more generous. All three were close to 70 pounds each. It probably would have cost me an extra $200 or more to check them if we had coach seats, but then we used the RCI luggage valet service so we never saw the bags from when we left the ship until we got home. I wonder how the airlines handle surcharges for that? Hold our bags for ransom at the end of the flights?
We've never had a problem with storage in the cabins though. Nesting the bags as Sonny does works for us too.