310 cadets are known for having different chamber lengths - and normally the BSA is the one with the adjustable rear sight, the military style ladder rear  was used by Greener.   
As the case uses a heeled bullet, case length is what controls 'seating depth', so it is a good idea to have dedicated cases for each rifle, trimmer to the right length.
However, they are pretty accurate indeed if your eyes can still manage the open sights.  Also, it is very easy to reload:
- Decap fired case
- Seat primer
- Drop powder
- Seat lubed bullet

That's it.  32-20 cases can be trimmed to be used, and you definitely need the heeled bullet - a .311 bullet definitely won't shoot well.

Are you sure the rifle that  fired the case in the middle was not rechambered to 32-20? That was a common thing to do.
Lee

My original Cadet does not need the donor 32-20 rims thinned down. Some do, some don't.
Fred

This is listed as a poll. What is the question?
Fred

The 310 uses an outside lubricated bullet of .316" diameter. Hence, the extra length of the 32-20 case just takes up the space normally occupied by the bullet.
If you have a Cadet that does not need the 32-20 rims thinned down, then ye:- it will probably chamber and fire the 32-20 round. Mind you, you won't be able to hit anything with it!
Fred