1. Actively managed funds diversified across equitieis, bonds, cash, etc
2. Index funds for specific indices, such as "top 4,000 US companies"
The danger with 2, they say, is you're not diversified. That'll be up to me I suppose to try to diversity into other areas.
The real lack of diversity is that it's weighted towards US companies which some people have decided they want and others don't, and apparently it's weighted towards stocks as an asset class as well.
I don't think weighting is the same as lacking diversification, although certainly you are lacking one kind of diversification.
Or, if one is of the "stocks don't go up over the long term" persuasion, it means stocks + shorts (for a net market neutral portfolio) and bonds.
So no, popular index funds are not diversified. There are exceptions, of course.