Many institutions are attracted to diversified portfolios of hedge funds, referred to as Funds of Hedge Funds (FoHFs). In this article, we examine a new database that separates out for the first time the effects of diversification (the number of underlying hedge funds) from scale (the magnitude of assets under management). We find with others that the variance-reducing effects of diversification diminish once FoHFs hold more than 20 underlying hedge funds. This excess diversification actually increases their left-tail risk exposure once we account for return smoothing. Furthermore, the average FoHF in our sample is more exposed to left-tail risk than are naive $1/N$ randomly chosen portfolios. This increase in tail risk is accompanied by lower returns, which we attribute to the cost of necessary due diligence that increases with the number of hedge funds.