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I have a 1291 with pistons, and for some reason my 3rd valve sticks once in a while. When I first purchased the tuba almost half a year ago, that same valve would stick very often. After having it chem-cleaned, everything was perfect. I regularly oil the pistons, so it came as a surprise that today it would stick. I removed the piston and oiled every single part of it, including the inside of the case. Without the valve cap, the piston went in smoothly, but whenever the cap was back on and I began playing, it still stuck all the way down. After more than an hour of pulling the piston in and out, the problem was finally fixed.

Could anybody give me some possible theories on why this particular valve tends to stick while no others ever do, so that I can be well prepared in the future. Thanks!

Check the braces on the valve tubing very carefully. I met a guy with a brand-new King a few years ago with a sometimes-sticky valve. We found that one brace was not soldered, so the tubing and valve cylinder flexed just a hair, causing the valve to sieze up when the instrument was being played. Same thing happened to a friend's cornet.
(There are many other possible causes which will be discussed here by other contributors.)

If it is consistently sticking in the down position as you seem to be describing that strongly suggests a bent valve stem. As you depress the button does the stem stay centered? If it does not, take the horn to a repair shop. Should be a quick and easy fix if that's the only issue.

Bloke's suggestion seems a likely cause as well. Honestly, you'll be much happier with your horn if you just make it a regular commitment to have your horn professionally cleaned once a year. We as tuba players have a bit of an attitude when it comes to paying someone else to clean our horns, but it matters.

Good luck