Acknowledgements: Hat tip to my colleague Jerold for stumbling upon this fix.
We had a user come in with a MacBook Air whose keyboard backlight wouldn't turn on. If you pressed the F5 or F6 key to turn it up or down, there would be on-screen feedback but instead of a indicator bar showing it going up or down, there would be only a "do not enter" sign appearing.
We tried an SMC reset and an NVRAM reset, to no avail.
Finally, Jerold just thought "Why don't we plug it into an external monitor?" For some reason, that fixed it (even after unplugging it from the external monitor).
So if you run into this issue, that may not be a sure fix, but it's certainly something to try!