I can't find Snapchat anywhere on the US stock exchanges as a public company, so I'm assuming that the Fidelity Blue Chip was holding this through the start up market. Are mutual funds allowed to hold companies that are off the US public markets?
Bloomberg suggests that two Fidelity funds hold preferred shares of Snapchat Inc.. Preferred shares hold more in common with bonds than with ordinary stock as they pay a fixed dividend, have lower liquidity, and don't have voting rights. Because of this lower liquidity they are not usually offered for sale on the market.
Whether these funds are allowed to hold such illiquid assets is more a question for their strategy document than the law; it is completely legal for a company to hold a non-marketable interest in another, even if the company is privately held as Snapchat is. The strategy documents governing what the fund is permitted to hold, however, may restrict ownership either banning non-market holdings or restricting the percentage of assets held in illiquid instruments.
Since IPO is very costly, funds like these who look to invest in new companies who have not been through IPO yet are a very good way of taking a diversified position in start-ups. Since they look to invest directly rather than through the market they are an attractive, low cost way for start-ups to generate funds to grow. The fund deals directly with the owners of the company to buy its shares.
The markdown of the stock value reflects the accounting principle of marking to market (MTM) financial assets that do not have a trade price so as to reflect their fair value. This markdown implies that Fidelity believe that the total NPV of the company's net assets is lower than they had previously calculated. This probably reflects a lack of revenue streams coming into the business in the case of Snapchat.
edit: by the way, since there is no market for start-up "stocks" pre-IPO my heart sinks a little every time I read the title of this question. I'm going to be sad all day now :(.