Most often they stop bubbling, but not always. Charles Delbeek, who heads up the Aquarium in Honolulu observed that the bubbling was related to the amount of light (in his case sunlight) shining on the anemone. I've read some articles that speculate that it has to do with whether the anemone is hungry or not, or it relates to water flow. I've observed in my tanks that one anemone might have bubble tips and be right beside another one that doesn't--and these are genetically identical clones.