I actually have no firm info - from the colors used to tile and the look of the rest of the bathroom (and because I know the family living here at the time did a lot of updating), I would say mid-80s. It does look later than the popcorn era, and later than asbestos, but as you said, still wouldn't hurt to check it.

I am SO glad you posted this, I have the same texture in a bathroom, of all places! I thought it was called popcorn, because I hadn't found any other good matches when I searched it. I should still get it checked for asbestos, I guess, but good to know it's easier to get rid of than popcorn if it's not.