Network theory proved recently to be useful in the quantification of many properties of financial systems. The analysis of the structure of investment portfolios is a major application since their eventual correlation and overlap impact the actual risk by individual investors. We investigate the bipartite network of US mutual fund portfolios and their assets. We follow its evolution during the Global Financial Crisis and study the diversification, as understood in modern portfolio theory, and the similarity of the investments of different funds. We show that, on average, portfolios have become more diversified and less similar during the crisis. However, we also find that large overlap is far more likely than expected from benchmark models of random allocation of investments. This indicates the existence of strong correlations between fund investment strategies. We exploit a deliberately simplified model of shock propagation to identify a systemic risk component stemming from the similarity of portfolios. The network is still partially vulnerable after the crisis because of this effect, despite the increase in the diversification of multi asset portfolios. Diversification and similarity should be taken into account jointly to properly assess systemic risk.