Too many fixed effects?

I haven’t seen this discussed before, but I think it could he quite important. Is there a downside to include too many fixed-effects in your regression? I am thinking about a diff-in-diff model in which I would include individual and establishment fixed-effects. If I were to include another FE (e.g. industry-year FE), is there a downside to this?

I somewhat recall hearing that an issue with including too many FE is that it may remove too much of the variation you’re interested in, but I may be mistaken here. Any insights here would be appreciated.