Can Cyprus happen in the United States? The short answer is: It can and it did.
What happened in Cyprus was that after wiping out the shareholders and debtholders of two failed banks (Laiki and Bank of Cyprus), there was still a deficit. This deficit was covered by having uninsured depositors taking on some losses (at present, the losses are said to be up to 62.5% for deposits over 100k Euros).
So why do I say the same thing can happen in the U.S.? It's simple. Because the standard procedure for closing down a failed bank in the U.S. is exactly the same. As it says in the FDIC website:
When a financial institution is closed and the Federal Deposit Insurance Corporation ("FDIC") is appointed as receiver, one of FDIC's responsibilities is to sell the institution's assets to pay the depositors and its creditors.
So what's the standard procedure when a bank fails? The FDIC comes in and the first thing it does is to move the insured deposits to another banking institution, so that they're available for depositors the very next day. The second thing is the FDIC wipes out shareholders, junior debtholders, senior debtholders and, if not enough assets are available, it then gives an haircut to uninsured depositors - that is, depositors whose deposits are over the FDIC's insurance threshold ($250k right now). In short, the same thing which happened in Cyprus can happen in the U.S.
But it's not just that the same thing happen can happen. It has already happened. It's simple - the standard operating procedure described above has already been applied multiple times in the past.
For instance, in one of the most famous failed institutions, IndyMac, there were about 8700 depositors who had deposits over the FDIC insurance threshold ($100k at the time). These 8700 depositors thus lost $266 million of their deposits, which constituted a loss of 50% over the insurance limit. This was basically the same as what happened in Cyprus, confirming that the same thing happened in the U.S. back in 2008.
The first exception
However, since late 2008 something else also happened. Apparently FDIC has been trying its best to sell the entirety of the deposits from closed institutions. Whereas up until then the practice was to move just the insured deposits, since then the FDIC has routinely been able to sell all deposits, in effect insulating uninsured deposits from losses. This change in policy is seemingly based on an unwritten rule, though, so nothing guarantees it will continue to be practiced.
The second exception - TBTF
There was also another deviation from standard operating procedure. A large number of banks during the 2008-2009 debt crisis, Citigroup (C) and Bank of America (BAC) chief among them, were treated as "TBTF" (Too Big To Fail). The major implication of this was that the U.S. Government and the Federal Reserve took exceptional measures, which allowed for public assistance (such as TARP) without first wiping out shareholders or debtholders. These exceptional measures also meant that uninsured deposits at these institutions, which could otherwise have suffered haircuts, took no losses at all.
Since this assistance was arbitrary, there's also no guarantee that it will be repeated if any financial institution again faces closure.
What happened in Cyprus was actually standard operating procedure for closing a failed financial institution in the United States up until a few years ago (2008). After the debt crisis took place, however, an unwritten policy went into effect, where small institutions regularly saw all their uninsured deposits saved by the FDIC through sales to acquiring banks, whereas large banks saw their uninsured deposits saved by bailouts on the grounds of those institutions being too big to fail.
As the situation normalizes (if we can call it that, with all the Fed printing), there's no guarantee that the standard operating procedure for closing failed institutions doesn't make a comeback, though.