Sounds like the one breaker has a direct short. Maybe someone deactivated a circuit and wire nutted the hot and neutral together.
This is a just a wild guess, but you've had some strange problems. You don't suppose those two breakers have a common trip and someone removed the handle tie so the circuits would be separate. The only way you would know is to remove the panel cover and see if the breakers are separate. If you do this, turn off the main breaker. The breakers should not be riveted together. You should be able to slip a piece thin cardboard or a playing card between them.
You have a 240V ( line to line) short with these two breakers. Somewhere down the line you have a homerun ( wire straight from the panel) that is connected or touching another phase from your panel. If you look at the breakers, one probably has a black wire, and the other probably has a red. ( hoping this is not a really old house with old wire) The best way of figuring this out is to see what is on the breaker that is always on. Then open the box, (outlet or light that is run off the working breaker) that is closest to the panel ( hopefully) and see if you have a cable that has three conductors ( plus ground). If you do, somewhere down the line the black and red are getting connectivity. You will have to trace it throghout the circuit and see where this is happening. Good luck.