Generally, consumers are free to invest in no matter what asset they are interested in. However consumers need to be either wealthy enough or have limits on the amount they can invest in some specific set of assets, such as hedge funds.
This is odd is we think that consumers could, for instance, buy a house and borrow heavily from a bank against the current value of the house, even if it is overvalued (as many are nowadays). Banks do not seem to worry much about whether market valuations are correct in relation to historical levels or some valuation model at the time of lending money against overvalued collateral. Consumers could invest as much money as they wanted in firms like Tyco, Worldcom, Enron