What Andy says is true, and additionally, if an out of spec synchromesh ring (specifically, if it's too thin, or the inner conical surface too large a diameter) it too can allow the sliding sychromesh hub__actuated by the shift fork itself__to over-travel, releasing the detent balls and trapping the springs, locking the xmsn into 4th gear. I was a victim of that very scenario during a trip from Louisiana to Atlanta, then to Florida sometime back in the mid-80s. I had rebuilt my 100/6 side-shift trans (same as your BN2) just prior to the trip, and noticed that I could "beat" the synchromesh if I did a quick and/or hard shift to 4th; I really didn't think too much of it at the time, and put it down to the quality of the then available reproduced parts I had just fitted.
On the road south of Atlanta, I made one of those quick/hard shifts to 4th, and got a brief cloud of smoke up through the cockpit. It was winter-time, so I was traveling with the top and side curtains erected. I pulled off the road, but couldn't shift the xmsn out of 4th gear__no matter how hard I pulled on the lever! I made the rest of the way to Spring Hill, FL using only 4th & 4th/ovd (I retarded the timing to ease the pinging when setting off from a stop). Kind of amazing really, how well I was still able to pull away from traffic from a dead stop!
Once I reached my brother's house (and garage and tools) I set about to pull the gearbox out. So not only could he not park his MGB in the garage when he got home from work, he had to help me man-handle the Healey's trans out of the car! I had drained the oil and pulled the side cover before committing to pull the trans, and the problem was easily identified, with mangled springs trapping the 3rd/4th synchromesh hub onto 4th gear.
A trip the next morning to the local Ace Hdwr for a tube of silicone sealant and a $0.35 assortment of springs (to cut/trim to length) to replace the damaged ones, and I could set about to reassemble the gearbox. I swapped the synchro rings from 3rd to 4th, preventing the same occurrence on the trip leg from Florida back to Louisiana.
Once back home, I ordered more synchro rings, balls, springs and another gasket set, then found the time to do it all over again. Nowadays, I check for things like that very carefully!
There are several albums on this gallery page relating to the rebuilding of a side-shift xmsn, if they'll help you with reference:
https://www.spcarsplus.com/gallery3/index.php/Healey-at-home/gearbox_work