Wouldn't it be great if this question was simply answered with "Yes!"?
With regard to your checking and savings deposits at FDIC-insured banks, it can be. You can trust that the money will be there even if the bank isn't. The only caveat is that the maximum amount covered by FDIC insurance is $250,000. Stay under that figure by keeping money in different banks, not a different branch of the same bank, and you can trust that you will get your money back.
But trust goes beyond just knowing your cash will be there when you need it. Recently, a topic of discussion with regulators has been how to best prevent the financial crisis from happening again. But enacting new legislation isn't likely to be easy, and many financial firms have already started to lobby heavily for their version of change. On the surface this may sound reasonable, as it will allow for profits to be less encumbered then they might be otherwise. (Read more in Are Your Bank Deposits Insured?)
This brings up an issue that many people know, but may not keep in mind: banks are profit-making institutions. Banks are borrowing money from you and then lending it out to others; they take the loan payment, pay you your interest, and keep the difference. So their incentive is to pay you the least amount they can, and to charge the most they can for loans, which may also be to you. (Find out how to get the bank to pay you for using their services, not the other way around.
Read Cut Your Bank Fees.)
Trust Banks to Make Profit
Furthermore, banks were allowed to combine with investment firms and insurance companies in 1999, due to the Financial Modernization Act. So, it's also in a bank's best interest to sell you a variety of products, some of which you may not really need, at the highest price possible. This is true for all profit-making firms, but for some reason many people lose sight of this incentive with banks. Perhaps it stems from the idea that you can trust banks to give you back the money you deposited. But it's clear that you can also trust banks to act like profit-making firms and do what's in their own interest.
So, "Yes!", you can trust your bank, in many different ways.