During the whole time in Cuba, I stuck to bottled water, even for brushing my teeth, and I drank mojitos and daiquiris and other iced beverages in more upscale restaurants and paladars without any trouble.
However, on the last night, the two halves of our group reunited for a farewell dinner in a more downscale restaurant. I ordered one of the set meals, and my neighbor to the left ordered the same one. The difference was that she ordered a beer, and I decided to have one last mojito.
The reaction began at midnight (sudden onset vomiting) and continued for about 16 hours, including the trip home.
So yes, it's true what they say: you can get sick from the ice in an alcoholic beverage. But if you're in a tourist restaurant, the kind that takes only CUCs, you should be OK.