It is true that this is possible, however, it's very remote in the case of the large and reputable fund companies such as Vanguard.
FDIC insurance protects against precisely this for bank accounts, but mutual funds and ETFs do not have an equivalent to FDIC insurance.
One thing that does help you in the case of a mutual fund or ETF is that you indirectly (through the fund) own actual assets. In a cash account at a bank, you have a promise from the bank to pay, and then the bank can go off and use your money to make loans. You don't in any sense own the bank's loans. With a fund, the fund company cannot (legally) take your money out of the fund, except to pay the expense ratio. They have to use your money to buy stocks, bonds, or whatever the fund invests in. Those assets are then owned by the fund.
Legally, a mutual fund is a special kind of company defined in the Investment Company Act of 1940, and is a separate company from the investment advisor (such as Vanguard):
Funds have their own boards, and in principle a fund board can even fire the company advising the fund, though this is not likely since boards aren't usually independent. (a quick google found this article for more, maybe someone can find a better one: http://www.marketwatch.com/story/mutual-fund-independent-board-rule-all-but-dead)
If Vanguard goes under, the funds could continue to exist and get a new adviser, or could be liquidated with investors receiving whatever the assets are worth.
Of course, all this legal stuff doesn't help you with outright fraud. If a fund's adviser says it bought the S&P 500, but really some guy bought himself a yacht, Madoff-style, then you have a problem. But a huge well-known ETF has auditors, tons of different employees, lots of brokerage and exchange traffic, etc. so to me at least it's tough to imagine a risk here. With a small fund company with just a few people - and there are lots of these! - then there's more risk, and you'd want to carefully look at what independent agent holds their assets, who their auditors are, and so forth.
With regular mutual funds (not ETFs) there are more issues with diversifying across fund companies:
- becomes tough to rebalance
- you may not qualify for lower expenses and fees (Vanguard and probably many others lower your fees if you have more assets with them)
- more risk of someone breaking in to your online accounts just because you have more
- increased complexity is never good, since it leads to procrastinating and less clarity
With ETFs, there probably isn't much downside to diversifying since you could buy them all from one brokerage account. Maybe it even happens naturally if you pick the best ETFs you can find. Personally, I would just pick the best ETFs and not worry about advisor diversity.
Update: maybe also deserving a mention are exchange-traded notes (ETNs). An ETN's legal structure is more like the bank account, minus the FDIC insurance of course. It's an IOU from the company that runs the ETN, where they promise to pay back the value of some index. There's no investment company as with a fund, and therefore you don't own a share of any actual assets. If the ETN's sponsor went bankrupt, you would indeed have a problem, much more so than if an ETF's sponsor went bankrupt.