Ooh, something I can talk about from experience. I have a consumer-grade FDM printer -- that's the most common type; it works by squeezing out plastic filament layer by layer. Think glue gun attached to an etch-a-sketch, controlled by a computer. You're mainly limited to ABS (same plastic as legos) or PLA (like compostable bioplastic food containers). I really wanted to be able to do the kinds of things being talked about on this thread, and my general conclusion is that it isn't worth it for more than temporary fixes. This is just my perspective as a hobbyist. Pros:
You can quickly test out the general size and shape of something you designed. Which is its own kind of satisfying, even if your object is ultimately unusable.
For cheaper Scholer- or Hohner-type concertinas, which are larger and have more plastic components anyway, printing replacement parts is more realistic. I was able to print a couple replacement buttons for the Scholer I started out on, but those were large-ish, uncomplicated plastic cylinders that only had to be epoxy'd onto the wooden action.
for the bits of the concertina that don't have anything to do with sound, printing can be a lifesaver in dire circumstances. E.g., I broke my wrist last year at the beginning of Morris season. I designed and printed a custom hand bar replacement that fit the shape of my cast and prevented it from sliding around, thus enabling me to at least play melodies with my left hand. To be fair, I could've just carved something out of wood, but hey, I didn't have great use of my dominant hand. Also ended up printing a replacement handle for someone's case.
Print resolution has been prohibitive. It isn't realistic with an FDM printer to do appropriately-sized buttons for standard instruments with riveted actions.
(Lack of) durability. Depending on your printer, your mileage may vary. If you print at really slow speeds and use as tiny a layer height as you can, you might get something usable, but it will be fragile.
Ridges. Unless you go nuts with post-processing, you're always going to have the ridges characteristic of all FDM-printed objects, which may bother you if your fingertips are in contact with the printed object for long periods of time.
Build volume. Though this is improving, most consumer printers are limited in terms of the size of object they can print in one piece. I haven't tried to do anything big like fretted ends or reedpans (not interested in that sound), but most consumer printers are going to have to do them in two pieces. Edit: RAc seems to have had good results with Shapeways, which is a way you can get around this limitation, but they get expensive (of course, buying a printer is expensive too).
I should say, if you splurge for one of the "prosumer" SLA printers that can do dental-grade prints ($5,000 and up), you'll have a different experience, but it'll still be plastic. Hope for the future: I wouldn't say that it's absolutely inappropriate to experiment with plastic instruments, long as it's acknowledged that they're imperfect prototypes. There's even a fully-printable violin that anyone can download. Obviously being demonstrated here by a pro, but it's far from the worst thing I've ever heard. Some people have gotten good results using a specialty filament that's partially made of wood, so that's interesting too. TL;DR: I wouldn't buy a 3D printer thinking it's going to be useful for concertina repair, but they're their own kind of fun.