Many, if not most US made bowlbacks had a neck/neckblock joint of some kind-I've mostly seen dovetails, but there probably are others. Many if not most Italian made bowlbacks have the neckblock integrally carved as part of the neck-though there may be other versions. #With this design, if the neck moves, rotates, breaks etc. etc. at this junction the repair consequences are more complicated-this may be the source of the 'deathknell' rumors you've heard.
As Paul indicates, removing the fretboard will reveal the joint-on most US bowlbacks, including Martin. # Is the joint loose? #Might you be able to steam apart the neck/neck block joint, clean it up and reglue it? #(It is tricky but easier than it might sound.) #I've done it on a number of old L+H and Washburn models to get my chops down. With some of the thicker fingerboards the whole deal is a slice of pie. Cosmetic precautions and touch up can outweigh the structural rigamarole.
Post some pictures if you can, there are enough folks around here to talk you through it. Maybe even Tappet Bros. it a bit. I agree, a Martin bowlback is quite worth the effort. I've got a modest 1907 bb that sounds and plays great. I have to keep myself from licking the rosewood. #Needed a bit of chiropractory of its own, btw.
Ever tried, ever failed, no matter. Try again, fail again, fail better.--Samuel Beckett