There are a variety of responses to such an offer. You can try to get it in writing, to address the risks associated with enforcing a verbal contract and while writing things down is always a really good idea, the really great part of writing things down is not that you've improved your odds in court which is never where you want to be (unless, I suppose, you are being paid to be there as someone's lawyer, or the bailiff or judge or person recording it or whatever, in which case you have a Job and that is presumably Good). The great part of writing things down is that it forces everyone to actually think about the details of what they are agreeing to. When the writing-things-down part is turned into pages and pages of small type which you cannot adjust to reflect verbal negotiations, things have probably already started to breakdown, unless you are buying a Known Quantity from a Trusted Seller, in which case, who really cares?
Problems can arise if we get a little careless with the multiple pages of small type that everyone ignores, especially if we unwittingly wander in Less-Well-Known Quantity territory, and, worse, with an Unknown Seller.
For example, we are all accustomed to putting our money in banks (for suitable definitions of "we", "all" and "banks"), and we expect that we will get our money back, even if something awful happens and the bank goes under. When we weren't so sure about this, we were much more likely to go take all our money out of the bank (believe it or not, doing that is "demanding" our "loan" be paid back immediately -- deposits are loans, and other than time deposits like CDs, they are callable) which increased the fragility of the banking system. In order to keep things nice and orderly and predictable, we removed that piece of uncertainty from the minds of depositors, aka, lenders.
Alas, when lenders quit worrying about capacity to repay, they may loan unwisely in search of high interest rates or some other benefit promised by the person who wants to borrow the money/bank which wants you to deposit money in it. There's a moral hazard/tradeoff.
Iceland and Cyprus are very, very different places, but in both cases, the national guarantor of banking deposits was too small to fulfill on its commitments. That is, anyone depositing money in Icelandic or Cypriot banks was relying upon a guarantee from someone who lacked the capacity to deliver.
"Honey, I'd love to give you double your money, but I got nothin'." *shows empty pockets*
Should people who were putting money in Icelandic and Cypriot banks have thought of this ahead of time? That's a great question. If individual depositors in a bank are thinking about whether the deposit guarantee is valid or not, then individual depositors may become another weak element in a chain, another part of the flood for the exits when a financial system experiences a shock or displacement or wtf. We Do Not Want That. It's so bad, that you can't really do _anything_ if ordinary depositors are prone to runs.
Iceland's native depositors were "made good", however, British and Dutch depositors were repeatedly excluded from legislation (and these weren't tax evaders -- these were people using an online bank because it was convenient and cool). In the end, the British and Dutch governments made their people good, and then took the whole thing to bankruptcy court to get first in line for payment and they will get almost all if not actually all of the money back. Thus, the ripple across the complacency of individual depositors was prevented from becoming something destabilizingly disastrous.
Cyprus' legislature, oddly, attempted to protect its external depositors at the expense of its internal depositors. I suspect they did this because they needed to be seen to make the gesture, however, it's possible they Really Are That Sold Out. In the end, the German taxpayers and other citizens of the EU will be paying the cost of making Cypriot depositors below a certain level whole, and amounts above that level will lose about half of what they had. Of course, without a bailout, things would have likely been much, much worse for Cypriot depositors of all sorts.
The chattering classes really went out and worked the "don't scare the depositors" meme for a few days, and then they split in two, with some saying, "it's all good you can quit worrying" and the rest going, "oooh, capital controls. They are horrible". And there was the usual, unseemly, European Monetary Union was/is a bad idea!
But operationally, the European Union is, with all deliberate speed, doing what it should be doing. It is going, hey, we cannot leave it to the member states to provide deposit insurance AND allow national banks to grow their balance sheets with wholesale funding.
Honestly, it is at this point we should all find employees of Moody's and shake our fingers at them. _They_ are the ones who rated these banks as safe to give job lots of money to at low interest rates, which is what enabled them to grow such a ridiculously large banking system in the first place. Because the legislatures of Iceland and Cyprus, however shockingly incompetent they may be (both in letting this happen in the first place, and then refusing to cooperate in their own salvation, stoking the flames of public anger instead), are democratically elected, it does seem wrong to just root them out and replace them with a technocratic body.
But you can sort of see how people might be tempted, while at the same time recognizing what a terrible idea that is.
So here we are, in a world filled with paranoia and blame, but also in a world in which people are _leaving their money in banks_, because they believe they'll be able to get that money later, if they need it. They're even okay with that in Cyprus. This is a necessary thing, because no edifice of debt, whether stable and of high quality or teetering and filled with fraud, can be built on anything other than the balance sheet typically made up of the savings and working money of ordinary human beings doing ordinary things.
Don't worry about the guarantor of the money in your bank, as long as your bank is in your country and they have a deposit guarantee and your deposit is within the limits of that guarantee. You have better things to do with your time and energy and other resources, like worrying about your job or your kids or your car.
But think long and hard before relying upon the deposit guarantee of a country other than your own, especially when you're putting your money there for reasons that do not bear scrutiny.