Just to give the OP nightmares: perhaps there is an open splice on a traveler in a junction box somewhere between the switches. 
Not to sound snarky, just to point out the obvious:
With the same diagram, if switch B is down, switch A is able to control the light and if switch B is up, the light is off no matter what. This would lead one to suspect an open contact in switch B.
Or better yet, just loose inside the wall in a stud bay. Buying a used house from a DIY idjit is always a voyage of discoveries. Mostly bad discoveries.
Here’s hoping it’s just a craptacular contractor-grade switch that’s run its course.
Our kitchen was done by someone who really didn’t understand how to do tile. We had to replace the floor tile, but decided to just live with the granite countertop tiles after getting a ridiculous estimate on replacement.
If the house is old enough that the switches are failing, replace both now so you don’t have this issue again in the near future.