Is it allowed? Practical? Stupid? Leaving aside for the moment the obvious issues of theft, etc., if I wanted to leave my roll-aboard behind for several days while bopping around with just my backpack, is this something I could conceivably consider? Or would it be either monumentally idiotic or just plain not allowed at all? If not both!
You've got a couple of options. Left luggage at the train station/airport (daily fee) or leaving your roll-aboard at a B&B, small hotel that you have stayed at for a few days and are planning to return to for at least one night. Done this many times with no problems. You can't expect a B&B/hotel etc. to store your luggage for free if there's nothing in it for them. I've stayed at a B&B in Rome for 4 nights, was gone 3 weeks, returned for 2 nights and my small suitcase was fine.
We too have left our large bags behind (traveling for 4 months) and taken a budget flight to somewhere else with just a backpack. The hotel was very accommodating and everything was waiting for us in "our room" when we returned from our jaunt. We stayed an additional 2 nights before flying home.
You cannot use locker as those generally have a 24 hr limit. Left luggage with an attendant will vary but my experience in a couple major cities (Rome in particular) is that they discourage long term storage by substantially increasing the daily rate so that it becomes very expensive, Of course, some will suggest that you should only have a backpack size in the first place. A hotel would be the best option.
We left our extra suitcase (clothes for wedding in Madrid) at our hotel for 2 weeks while we drove around Spain. The bag was fine and there was no extra charge, but we stayed at the hotel at the beginning and the end of this trip. I prefer this method to leaving bag(s) in a public station locker for a long period of time.
I've done it before. I added a week at the end of a business trip for touring. I ped my business stuff at Frankfurt's left luggage and traveled with only an extra pair of pants, two extra shirts, and two changes of underwear and socks in a daypack. I liked it and it was no problem at all. Frankfurts left luggage is checked in, like a coat check, and x-rayed. Getting a locker at a train station is probably easier, and cheaper. The last one I used was Copenhagen and their system was a little different than most. Make sure you know the rules on how long items can remain there so you don't learn any hard lessons.